src/HOL/Library/Topology_Euclidean_Space.thy
author huffman
Mon, 08 Jun 2009 14:44:53 -0700
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parent 31529 689f5dae1f41
child 31531 fc78714d14e1
permissions -rw-r--r--
generalize constant 'indirection'
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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(* Title:      Topology
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   Author:     Amine Chaieb, University of Cambridge
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   Author:     Robert Himmelmann, TU Muenchen
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*)
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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header {* Elementary topology in Euclidean space. *}
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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theory Topology_Euclidean_Space
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254478a8dd05 dropped theory Arith_Tools
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imports SEQ Euclidean_Space
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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begin
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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declare fstcart_pastecart[simp] sndcart_pastecart[simp]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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subsection{* General notion of a topology *}
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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definition "istopology L \<longleftrightarrow> {} \<in> L \<and> (\<forall>S \<in>L. \<forall>T \<in>L. S \<inter> T \<in> L) \<and> (\<forall>K. K \<subseteq>L \<longrightarrow> \<Union> K \<in> L)"
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typedef (open) 'a topology = "{L::('a set) set. istopology L}"
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  morphisms "openin" "topology"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  unfolding istopology_def by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma istopology_open_in[intro]: "istopology(openin U)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  using openin[of U] by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  using topology_inverse[unfolded mem_def Collect_def] .
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
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  using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  {assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    hence "openin T1 = openin T2" by (metis mem_def set_ext)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    36
    hence "topology (openin T1) = topology (openin T2)" by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    hence "T1 = T2" unfolding openin_inverse .}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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  ultimately show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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text{* Infer the "universe" from union of all sets in the topology. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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definition "topspace T =  \<Union>{S. openin T S}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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subsection{* Main properties of open sets *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_clauses:
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  fixes U :: "'a topology"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  shows "openin U {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  using openin[of U] unfolding istopology_def Collect_def mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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  by (metis mem_def subset_eq)+
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  unfolding topspace_def by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  by (simp add: openin_clauses)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)" by (simp add: openin_clauses)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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parents:
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  using openin_Union[of "{S,T}" U] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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  {assume ?lhs then have ?rhs by auto }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
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    72
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
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    73
  {assume H: ?rhs
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    74
    then obtain t where t: "\<forall>x\<in>S. openin U (t x) \<and> x \<in> t x \<and> t x \<subseteq> S"
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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      unfolding Ball_def ex_simps(6)[symmetric] choice_iff by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    76
    from t have th0: "\<forall>x\<in> t`S. openin U x" by auto
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    77
    have "\<Union> t`S = S" using t by auto
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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parents:
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    78
    with openin_Union[OF th0] have "openin U S" by simp }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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parents:
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    79
  ultimately show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    80
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    81
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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subsection{* Closed sets *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    84
definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    85
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    86
lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    87
lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
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    88
lemma closedin_topspace[intro,simp]:
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    89
  "closedin U (topspace U)" by (simp add: closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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    90
lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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    91
  by (auto simp add: Diff_Un closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    92
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    93
lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    94
lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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    95
  shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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parents:
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    96
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
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    97
lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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    98
  using closedin_Inter[of "{S,T}" U] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    99
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   100
lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   101
lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   102
  apply (auto simp add: closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   103
  apply (metis openin_subset subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   104
  apply (auto simp add: Diff_Diff_Int)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   105
  apply (subgoal_tac "topspace U \<inter> S = S")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   106
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   107
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   108
lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   109
  by (simp add: openin_closedin_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   110
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
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   111
lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   112
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   113
  have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   114
    by (auto simp add: topspace_def openin_subset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   115
  then show ?thesis using oS cT by (auto simp add: closedin_def)
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huffman
parents: 30268
diff changeset
   116
qed
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5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   117
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   118
lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   119
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   120
  have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   121
    by (auto simp add: topspace_def )
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   122
  then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   123
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   124
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   125
subsection{* Subspace topology. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   126
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   127
definition "subtopology U V = topology {S \<inter> V |S. openin U S}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   128
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   129
lemma istopology_subtopology: "istopology {S \<inter> V |S. openin U S}" (is "istopology ?L")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   130
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   131
  have "{} \<in> ?L" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   132
  {fix A B assume A: "A \<in> ?L" and B: "B \<in> ?L"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   133
    from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   134
    have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   135
    then have "A \<inter> B \<in> ?L" by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   136
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   137
  {fix K assume K: "K \<subseteq> ?L"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   138
    have th0: "?L = (\<lambda>S. S \<inter> V) ` openin U "
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   139
      apply (rule set_ext)
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   140
      apply (simp add: Ball_def image_iff)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   141
      by (metis mem_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   142
    from K[unfolded th0 subset_image_iff]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   143
    obtain Sk where Sk: "Sk \<subseteq> openin U" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   144
    have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   145
    moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq mem_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   146
    ultimately have "\<Union>K \<in> ?L" by blast}
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   147
  ultimately show ?thesis unfolding istopology_def by blast
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   148
qed
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   149
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   150
lemma openin_subtopology:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   151
  "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   152
  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   153
  by (auto simp add: Collect_def)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   154
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   155
lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   156
  by (auto simp add: topspace_def openin_subtopology)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   157
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   158
lemma closedin_subtopology:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   159
  "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   160
  unfolding closedin_def topspace_subtopology
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   161
  apply (simp add: openin_subtopology)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   162
  apply (rule iffI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   163
  apply clarify
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   164
  apply (rule_tac x="topspace U - T" in exI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   165
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   166
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   167
lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   168
  unfolding openin_subtopology
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   169
  apply (rule iffI, clarify)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   170
  apply (frule openin_subset[of U])  apply blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   171
  apply (rule exI[where x="topspace U"])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   172
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   173
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   174
lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   175
  shows "subtopology U V = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   176
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   177
  {fix S
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   178
    {fix T assume T: "openin U T" "S = T \<inter> V"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   179
      from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   180
      have "openin U S" unfolding eq using T by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   181
    moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   182
    {assume S: "openin U S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   183
      hence "\<exists>T. openin U T \<and> S = T \<inter> V"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   184
	using openin_subset[OF S] UV by auto}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   185
    ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   186
  then show ?thesis unfolding topology_eq openin_subtopology by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   187
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   188
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   189
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   190
lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   191
  by (simp add: subtopology_superset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   192
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   193
lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   194
  by (simp add: subtopology_superset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   195
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   196
subsection{* The universal Euclidean versions are what we use most of the time *}
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   197
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   198
definition
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   199
  euclidean :: "'a::topological_space topology" where
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   200
  "euclidean = topology open"
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   201
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   202
lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   203
  unfolding euclidean_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   204
  apply (rule cong[where x=S and y=S])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   205
  apply (rule topology_inverse[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   206
  apply (auto simp add: istopology_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   207
  by (auto simp add: mem_def subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   208
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   209
lemma topspace_euclidean: "topspace euclidean = UNIV"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   210
  apply (simp add: topspace_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   211
  apply (rule set_ext)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   212
  by (auto simp add: open_openin[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   213
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   214
lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   215
  by (simp add: topspace_euclidean topspace_subtopology)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   216
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   217
lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
31490
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31489
diff changeset
   218
  by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   219
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   220
lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   221
  by (simp add: open_openin openin_subopen[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   222
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   223
subsection{* Open and closed balls. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   224
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   225
definition
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   226
  ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   227
  "ball x e = {y. dist x y < e}"
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   228
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   229
definition
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   230
  cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   231
  "cball x e = {y. dist x y \<le> e}"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   232
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   233
lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   234
lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   235
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   236
lemma mem_ball_0 [simp]:
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   237
  fixes x :: "'a::real_normed_vector"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   238
  shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   239
  by (simp add: dist_norm)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   240
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   241
lemma mem_cball_0 [simp]:
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   242
  fixes x :: "'a::real_normed_vector"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   243
  shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   244
  by (simp add: dist_norm)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   245
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   246
lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e"  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   247
lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   248
lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   249
lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   250
lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   251
  by (simp add: expand_set_eq) arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   252
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   253
lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   254
  by (simp add: expand_set_eq)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   255
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   256
subsection{* Topological properties of open balls *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   257
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   258
lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   259
  "(a::real) - b < 0 \<longleftrightarrow> a < b"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   260
  "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   261
lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   262
  "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   263
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   264
lemma open_ball[intro, simp]: "open (ball x e)"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   265
  unfolding open_dist ball_def Collect_def Ball_def mem_def
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   266
  unfolding dist_commute
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   267
  apply clarify
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   268
  apply (rule_tac x="e - dist xa x" in exI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   269
  using dist_triangle_alt[where z=x]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   270
  apply (clarsimp simp add: diff_less_iff)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   271
  apply atomize
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   272
  apply (erule_tac x="y" in allE)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   273
  apply (erule_tac x="xa" in allE)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   274
  by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   275
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   276
lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   277
lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   278
  unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   279
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   280
lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   281
  by (metis open_contains_ball subset_eq centre_in_ball)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   282
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   283
lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   284
  unfolding mem_ball expand_set_eq
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   285
  apply (simp add: not_less)
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   286
  by (metis zero_le_dist order_trans dist_self)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   287
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   288
lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   289
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   290
subsection{* Basic "localization" results are handy for connectedness. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   291
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   292
lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   293
  by (auto simp add: openin_subtopology open_openin[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   294
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   295
lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   296
  by (auto simp add: openin_open)
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   297
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   298
lemma open_openin_trans[trans]:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   299
 "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   300
  by (metis Int_absorb1  openin_open_Int)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   301
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   302
lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   303
  by (auto simp add: openin_open)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   304
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   305
lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   306
  by (simp add: closedin_subtopology closed_closedin Int_ac)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   307
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   308
lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   309
  by (metis closedin_closed)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   310
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   311
lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   312
  apply (subgoal_tac "S \<inter> T = T" )
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   313
  apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   314
  apply (frule closedin_closed_Int[of T S])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   315
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   316
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   317
lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   318
  by (auto simp add: closedin_closed)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   319
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   320
lemma openin_euclidean_subtopology_iff:
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   321
  fixes S U :: "'a::metric_space set"
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   322
  shows "openin (subtopology euclidean U) S
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   323
  \<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   324
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   325
  {assume ?lhs hence ?rhs unfolding openin_subtopology open_openin[symmetric]
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   326
      by (simp add: open_dist) blast}
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   327
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   328
  {assume SU: "S \<subseteq> U" and H: "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x' \<in> S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   329
    from H obtain d where d: "\<And>x . x\<in> S \<Longrightarrow> d x > 0 \<and> (\<forall>x' \<in> U. dist x' x < d x \<longrightarrow> x' \<in> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   330
      by metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   331
    let ?T = "\<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   332
    have oT: "open ?T" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   333
    { fix x assume "x\<in>S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   334
      hence "x \<in> \<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   335
	apply simp apply(rule_tac x="ball x(d x)" in exI) apply auto
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   336
        by (rule d [THEN conjunct1])
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   337
      hence "x\<in> ?T \<inter> U" using SU and `x\<in>S` by auto  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   338
    moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   339
    { fix y assume "y\<in>?T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   340
      then obtain B where "y\<in>B" "B\<in>{B. \<exists>x\<in>S. B = ball x (d x)}" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   341
      then obtain x where "x\<in>S" and x:"y \<in> ball x (d x)" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   342
      assume "y\<in>U"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   343
      hence "y\<in>S" using d[OF `x\<in>S`] and x by(auto simp add: dist_commute) }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   344
    ultimately have "S = ?T \<inter> U" by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   345
    with oT have ?lhs unfolding openin_subtopology open_openin[symmetric] by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   346
  ultimately show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   347
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   348
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   349
text{* These "transitivity" results are handy too. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   350
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   351
lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   352
  \<Longrightarrow> openin (subtopology euclidean U) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   353
  unfolding open_openin openin_open by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   354
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   355
lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   356
  by (auto simp add: openin_open intro: openin_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   357
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   358
lemma closedin_trans[trans]:
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   359
 "closedin (subtopology euclidean T) S \<Longrightarrow>
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   360
           closedin (subtopology euclidean U) T
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   361
           ==> closedin (subtopology euclidean U) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   362
  by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   363
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   364
lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   365
  by (auto simp add: closedin_closed intro: closedin_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   366
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   367
subsection{* Connectedness *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   368
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   369
definition "connected S \<longleftrightarrow>
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   370
  ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   371
  \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   372
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   373
lemma connected_local:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   374
 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   375
                 openin (subtopology euclidean S) e1 \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   376
                 openin (subtopology euclidean S) e2 \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   377
                 S \<subseteq> e1 \<union> e2 \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   378
                 e1 \<inter> e2 = {} \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   379
                 ~(e1 = {}) \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   380
                 ~(e2 = {}))"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   381
unfolding connected_def openin_open by (safe, blast+)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   382
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   383
lemma exists_diff: "(\<exists>S. P(UNIV - S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   384
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   385
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   386
  {assume "?lhs" hence ?rhs by blast }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   387
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   388
  {fix S assume H: "P S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   389
    have "S = UNIV - (UNIV - S)" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   390
    with H have "P (UNIV - (UNIV - S))" by metis }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   391
  ultimately show ?thesis by metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   392
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   393
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   394
lemma connected_clopen: "connected S \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   395
        (\<forall>T. openin (subtopology euclidean S) T \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   396
            closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   397
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   398
  have " \<not> connected S \<longleftrightarrow> (\<exists>e1 e2. open e1 \<and> open (UNIV - e2) \<and> S \<subseteq> e1 \<union> (UNIV - e2) \<and> e1 \<inter> (UNIV - e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (UNIV - e2) \<inter> S \<noteq> {})"
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   399
    unfolding connected_def openin_open closedin_closed
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   400
    apply (subst exists_diff) by blast
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   401
  hence th0: "connected S \<longleftrightarrow> \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (UNIV - e2) \<and> e1 \<inter> (UNIV - e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (UNIV - e2) \<inter> S \<noteq> {})"
31490
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31489
diff changeset
   402
    (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def Compl_eq_Diff_UNIV) by metis
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   403
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   404
  have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   405
    (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   406
    unfolding connected_def openin_open closedin_closed by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   407
  {fix e2
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   408
    {fix e1 have "?P e2 e1 \<longleftrightarrow> (\<exists>t.  closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t\<noteq>S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   409
	by auto}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   410
    then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   411
  then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   412
  then show ?thesis unfolding th0 th1 by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   413
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   414
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   415
lemma connected_empty[simp, intro]: "connected {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   416
  by (simp add: connected_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   417
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   418
subsection{* Hausdorff and other separation properties *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   419
31457
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   420
class t0_space =
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   421
  assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   422
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   423
class t1_space =
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   424
  assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<notin> U \<and> x \<notin> V \<and> y \<in> V"
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   425
begin
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   426
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   427
subclass t0_space
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   428
proof
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   429
qed (fast dest: t1_space)
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   430
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   431
end
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   432
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   433
text {* T2 spaces are also known as Hausdorff spaces. *}
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   434
31457
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   435
class t2_space =
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   436
  assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   437
begin
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   438
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   439
subclass t1_space
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   440
proof
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   441
qed (fast dest: hausdorff)
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   442
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   443
end
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   444
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   445
instance metric_space \<subseteq> t2_space
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   446
proof
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   447
  fix x y :: "'a::metric_space"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   448
  assume xy: "x \<noteq> y"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   449
  let ?U = "ball x (dist x y / 2)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   450
  let ?V = "ball y (dist x y / 2)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   451
  have th0: "\<And>d x y z. (d x z :: real) <= d x y + d y z \<Longrightarrow> d y z = d z y
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   452
               ==> ~(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   453
  have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   454
    using dist_pos_lt[OF xy] th0[of dist,OF dist_triangle dist_commute]
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   455
    by (auto simp add: expand_set_eq)
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   456
  then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   457
    by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   458
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   459
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   460
lemma separation_t2:
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   461
  fixes x y :: "'a::t2_space"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   462
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   463
  using hausdorff[of x y] by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   464
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   465
lemma separation_t1:
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   466
  fixes x y :: "'a::t1_space"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   467
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in>U \<and> y\<notin> U \<and> x\<notin>V \<and> y\<in>V)"
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   468
  using t1_space[of x y] by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   469
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   470
lemma separation_t0:
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   471
  fixes x y :: "'a::t0_space"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   472
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   473
  using t0_space[of x y] by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   474
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   475
subsection{* Limit points *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   476
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   477
definition
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   478
  islimpt:: "'a::topological_space \<Rightarrow> 'a set \<Rightarrow> bool"
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   479
    (infixr "islimpt" 60) where
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   480
  "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   481
31489
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   482
lemma islimptI:
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   483
  assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   484
  shows "x islimpt S"
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   485
  using assms unfolding islimpt_def by auto
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   486
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   487
lemma islimptE:
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   488
  assumes "x islimpt S" and "x \<in> T" and "open T"
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   489
  obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   490
  using assms unfolding islimpt_def by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   491
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   492
lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   493
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   494
lemma islimpt_approachable:
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   495
  fixes x :: "'a::metric_space"
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   496
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   497
  unfolding islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   498
  apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   499
  apply(erule_tac x="ball x e" in allE)
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   500
  apply auto
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   501
  apply(rule_tac x=y in bexI)
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   502
  apply (auto simp add: dist_commute)
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   503
  apply (simp add: open_dist, drule (1) bspec)
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   504
  apply (clarify, drule spec, drule (1) mp, auto)
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   505
  done
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   506
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   507
lemma islimpt_approachable_le:
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   508
  fixes x :: "'a::metric_space"
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   509
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   510
  unfolding islimpt_approachable
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   511
  using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   512
  by metis (* FIXME: VERY slow! *)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   513
31457
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   514
class perfect_space =
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   515
  (* FIXME: perfect_space should inherit from topological_space *)
31457
d1cb222438d8 class replaces axclass
haftmann
parents: 31421
diff changeset
   516
  assumes islimpt_UNIV [simp, intro]: "(x::'a::metric_space) islimpt UNIV"
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   517
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   518
lemma perfect_choose_dist:
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   519
  fixes x :: "'a::perfect_space"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   520
  shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   521
using islimpt_UNIV [of x]
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   522
by (simp add: islimpt_approachable)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   523
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   524
instance real :: perfect_space
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   525
apply default
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   526
apply (rule islimpt_approachable [THEN iffD2])
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   527
apply (clarify, rule_tac x="x + e/2" in bexI)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   528
apply (auto simp add: dist_norm)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   529
done
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   530
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   531
instance "^" :: (perfect_space, finite) perfect_space
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   532
proof
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   533
  fix x :: "'a ^ 'b"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   534
  {
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   535
    fix e :: real assume "0 < e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   536
    def a \<equiv> "x $ arbitrary"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   537
    have "a islimpt UNIV" by (rule islimpt_UNIV)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   538
    with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   539
      unfolding islimpt_approachable by auto
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   540
    def y \<equiv> "Cart_lambda ((Cart_nth x)(arbitrary := b))"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   541
    from `b \<noteq> a` have "y \<noteq> x"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   542
      unfolding a_def y_def by (simp add: Cart_eq)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   543
    from `dist b a < e` have "dist y x < e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   544
      unfolding dist_vector_def a_def y_def
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   545
      apply simp
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   546
      apply (rule le_less_trans [OF setL2_le_setsum [OF zero_le_dist]])
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   547
      apply (subst setsum_diff1' [where a=arbitrary], simp, simp, simp)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   548
      done
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   549
    from `y \<noteq> x` and `dist y x < e`
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   550
    have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   551
  }
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   552
  then show "x islimpt UNIV" unfolding islimpt_approachable by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   553
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   554
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   555
lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   556
  unfolding closed_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   557
  apply (subst open_subopen)
31490
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31489
diff changeset
   558
  apply (simp add: islimpt_def subset_eq Compl_eq_Diff_UNIV)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   559
  by (metis DiffE DiffI UNIV_I insertCI insert_absorb mem_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   560
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   561
lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   562
  unfolding islimpt_def by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   563
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   564
lemma closed_positive_orthant: "closed {x::real^'n::finite. \<forall>i. 0 \<le>x$i}"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   565
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   566
  let ?U = "UNIV :: 'n set"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   567
  let ?O = "{x::real^'n. \<forall>i. x$i\<ge>0}"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   568
  {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   569
    and xi: "x$i < 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   570
    from xi have th0: "-x$i > 0" by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   571
    from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x $ i" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   572
      have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   573
      have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   574
      have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using x'(1) xi
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   575
	apply (simp only: vector_component)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   576
	by (rule th') auto
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   577
      have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using  component_le_norm[of "x'-x" i]
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31286
diff changeset
   578
	apply (simp add: dist_norm) by norm
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   579
      from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   580
  then show ?thesis unfolding closed_limpt islimpt_approachable
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   581
    unfolding not_le[symmetric] by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   582
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   583
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31286
diff changeset
   584
lemma finite_set_avoid:
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   585
  fixes a :: "'a::metric_space"
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31286
diff changeset
   586
  assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   587
proof(induct rule: finite_induct[OF fS])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   588
  case 1 thus ?case apply auto by ferrack
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   589
next
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   590
  case (2 x F)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   591
  from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   592
  {assume "x = a" hence ?case using d by auto  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   593
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   594
  {assume xa: "x\<noteq>a"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   595
    let ?d = "min d (dist a x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   596
    have dp: "?d > 0" using xa d(1) using dist_nz by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   597
    from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   598
    with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   599
  ultimately show ?case by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   600
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   601
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   602
lemma islimpt_finite:
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   603
  fixes S :: "'a::metric_space set"
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   604
  assumes fS: "finite S" shows "\<not> a islimpt S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   605
  unfolding islimpt_approachable
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   606
  using finite_set_avoid[OF fS, of a] by (metis dist_commute  not_le)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   607
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   608
lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   609
  apply (rule iffI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   610
  defer
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   611
  apply (metis Un_upper1 Un_upper2 islimpt_subset)
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   612
  unfolding islimpt_def
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   613
  apply (rule ccontr, clarsimp, rename_tac A B)
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   614
  apply (drule_tac x="A \<inter> B" in spec)
31490
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31489
diff changeset
   615
  apply (auto simp add: open_Int)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   616
  done
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   617
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   618
lemma discrete_imp_closed:
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   619
  fixes S :: "'a::metric_space set"
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   620
  assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   621
  shows "closed S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   622
proof-
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   623
  {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   624
    from e have e2: "e/2 > 0" by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   625
    from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   626
    let ?m = "min (e/2) (dist x y) "
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   627
    from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   628
    from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   629
    have th: "dist z y < e" using z y
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   630
      by (intro dist_triangle_lt [where z=x], simp)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   631
    from d[rule_format, OF y(1) z(1) th] y z
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   632
    have False by (auto simp add: dist_commute)}
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   633
  then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   634
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   635
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   636
subsection{* Interior of a Set *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   637
definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   638
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   639
lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   640
  apply (simp add: expand_set_eq interior_def)
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   641
  apply (subst (2) open_subopen) by (safe, blast+)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   642
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   643
lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   644
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   645
lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   646
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   647
lemma open_interior[simp, intro]: "open(interior S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   648
  apply (simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   649
  apply (subst open_subopen) by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   650
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   651
lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   652
lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   653
lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   654
lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   655
lemma interior_unique: "T \<subseteq> S \<Longrightarrow> open T  \<Longrightarrow> (\<forall>T'. T' \<subseteq> S \<and> open T' \<longrightarrow> T' \<subseteq> T) \<Longrightarrow> interior S = T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   656
  by (metis equalityI interior_maximal interior_subset open_interior)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   657
lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   658
  apply (simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   659
  by (metis open_contains_ball centre_in_ball open_ball subset_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   660
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   661
lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   662
  by (metis interior_maximal interior_subset subset_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   663
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   664
lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   665
  apply (rule equalityI, simp)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   666
  apply (metis Int_lower1 Int_lower2 subset_interior)
31490
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31489
diff changeset
   667
  by (metis Int_mono interior_subset open_Int open_interior open_subset_interior)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   668
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   669
lemma interior_limit_point [intro]:
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   670
  fixes x :: "'a::perfect_space"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   671
  assumes x: "x \<in> interior S" shows "x islimpt S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   672
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   673
  from x obtain e where e: "e>0" "\<forall>x'. dist x x' < e \<longrightarrow> x' \<in> S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   674
    unfolding mem_interior subset_eq Ball_def mem_ball by blast
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   675
  {
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   676
    fix d::real assume d: "d>0"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   677
    let ?m = "min d e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   678
    have mde2: "0 < ?m" using e(1) d(1) by simp
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   679
    from perfect_choose_dist [OF mde2, of x]
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   680
    obtain y where "y \<noteq> x" and "dist y x < ?m" by blast
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   681
    then have "dist y x < e" "dist y x < d" by simp_all
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   682
    from `dist y x < e` e(2) have "y \<in> S" by (simp add: dist_commute)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   683
    have "\<exists>x'\<in>S. x'\<noteq> x \<and> dist x' x < d"
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   684
      using `y \<in> S` `y \<noteq> x` `dist y x < d` by fast
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   685
  }
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   686
  then show ?thesis unfolding islimpt_approachable by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   687
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   688
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   689
lemma interior_closed_Un_empty_interior:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   690
  assumes cS: "closed S" and iT: "interior T = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   691
  shows "interior(S \<union> T) = interior S"
31394
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   692
proof
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   693
  show "interior S \<subseteq> interior (S\<union>T)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   694
    by (rule subset_interior, blast)
31394
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   695
next
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   696
  show "interior (S \<union> T) \<subseteq> interior S"
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   697
  proof
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   698
    fix x assume "x \<in> interior (S \<union> T)"
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   699
    then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T"
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   700
      unfolding interior_def by fast
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   701
    show "x \<in> interior S"
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   702
    proof (rule ccontr)
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   703
      assume "x \<notin> interior S"
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   704
      with `x \<in> R` `open R` obtain y where "y \<in> R - S"
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   705
        unfolding interior_def expand_set_eq by fast
31490
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31489
diff changeset
   706
      from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
31394
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   707
      from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   708
      from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   709
      show "False" unfolding interior_def by fast
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   710
    qed
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   711
  qed
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   712
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   713
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   714
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   715
subsection{* Closure of a Set *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   716
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   717
definition "closure S = S \<union> {x | x. x islimpt S}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   718
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   719
lemma closure_interior: "closure S = UNIV - interior (UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   720
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   721
  { fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   722
    have "x\<in>UNIV - interior (UNIV - S) \<longleftrightarrow> x \<in> closure S"  (is "?lhs = ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   723
    proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   724
      let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   725
      assume "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   726
      hence *:"\<not> ?exT x"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   727
	unfolding interior_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   728
	by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   729
      { assume "\<not> ?rhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   730
	hence False using *
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   731
	  unfolding closure_def islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   732
	  by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   733
      }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   734
      thus "?rhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   735
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   736
    next
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   737
      assume "?rhs" thus "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   738
	unfolding closure_def interior_def islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   739
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   740
    qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   741
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   742
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   743
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   744
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   745
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   746
lemma interior_closure: "interior S = UNIV - (closure (UNIV - S))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   747
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   748
  { fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   749
    have "x \<in> interior S \<longleftrightarrow> x \<in> UNIV - (closure (UNIV - S))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   750
      unfolding interior_def closure_def islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   751
      by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   752
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   753
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   754
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   755
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   756
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   757
lemma closed_closure[simp, intro]: "closed (closure S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   758
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   759
  have "closed (UNIV - interior (UNIV -S))" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   760
  thus ?thesis using closure_interior[of S] by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   761
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   762
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   763
lemma closure_hull: "closure S = closed hull S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   764
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   765
  have "S \<subseteq> closure S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   766
    unfolding closure_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   767
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   768
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   769
  have "closed (closure S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   770
    using closed_closure[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   771
    by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   772
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   773
  { fix t
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   774
    assume *:"S \<subseteq> t" "closed t"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   775
    { fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   776
      assume "x islimpt S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   777
      hence "x islimpt t" using *(1)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   778
	using islimpt_subset[of x, of S, of t]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   779
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   780
    }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   781
    with * have "closure S \<subseteq> t"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   782
      unfolding closure_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   783
      using closed_limpt[of t]
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   784
      by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   785
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   786
  ultimately show ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   787
    using hull_unique[of S, of "closure S", of closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   788
    unfolding mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   789
    by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   790
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   791
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   792
lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   793
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   794
  using hull_eq[of closed, unfolded mem_def, OF  closed_Inter, of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   795
  by (metis mem_def subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   796
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   797
lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   798
  using closure_eq[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   799
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   800
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   801
lemma closure_closure[simp]: "closure (closure S) = closure S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   802
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   803
  using hull_hull[of closed S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   804
  by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   805
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   806
lemma closure_subset: "S \<subseteq> closure S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   807
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   808
  using hull_subset[of S closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   809
  by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   810
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   811
lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   812
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   813
  using hull_mono[of S T closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   814
  by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   815
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   816
lemma closure_minimal: "S \<subseteq> T \<Longrightarrow>  closed T \<Longrightarrow> closure S \<subseteq> T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   817
  using hull_minimal[of S T closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   818
  unfolding closure_hull mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   819
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   820
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   821
lemma closure_unique: "S \<subseteq> T \<and> closed T \<and> (\<forall> T'. S \<subseteq> T' \<and> closed T' \<longrightarrow> T \<subseteq> T') \<Longrightarrow> closure S = T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   822
  using hull_unique[of S T closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   823
  unfolding closure_hull mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   824
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   825
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   826
lemma closure_empty[simp]: "closure {} = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   827
  using closed_empty closure_closed[of "{}"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   828
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   829
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   830
lemma closure_univ[simp]: "closure UNIV = UNIV"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   831
  using closure_closed[of UNIV]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   832
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   833
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   834
lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   835
  using closure_empty closure_subset[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   836
  by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   837
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   838
lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   839
  using closure_eq[of S] closure_subset[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   840
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   841
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   842
lemma open_inter_closure_eq_empty:
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   843
  "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   844
  using open_subset_interior[of S "UNIV - T"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   845
  using interior_subset[of "UNIV - T"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   846
  unfolding closure_interior
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   847
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   848
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   849
lemma open_inter_closure_subset:
31489
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   850
  "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   851
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   852
  fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   853
  assume as: "open S" "x \<in> S \<inter> closure T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   854
  { assume *:"x islimpt T"
31489
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   855
    have "x islimpt (S \<inter> T)"
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   856
    proof (rule islimptI)
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   857
      fix A
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   858
      assume "x \<in> A" "open A"
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   859
      with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
31490
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31489
diff changeset
   860
        by (simp_all add: open_Int)
31489
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   861
      with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   862
        by (rule islimptE)
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   863
      hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   864
        by simp_all
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   865
      thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
10080e31b294 lemmas islimptI, islimptE; generalize open_inter_closure_subset
huffman
parents: 31488
diff changeset
   866
    qed
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   867
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   868
  then show "x \<in> closure (S \<inter> T)" using as
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   869
    unfolding closure_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   870
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   871
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   872
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   873
lemma closure_complement: "closure(UNIV - S) = UNIV - interior(S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   874
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   875
  have "S = UNIV - (UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   876
    by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   877
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   878
    unfolding closure_interior
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   879
    by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   880
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   881
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   882
lemma interior_complement: "interior(UNIV - S) = UNIV - closure(S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   883
  unfolding closure_interior
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   884
  by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   885
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   886
subsection{* Frontier (aka boundary) *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   887
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   888
definition "frontier S = closure S - interior S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   889
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   890
lemma frontier_closed: "closed(frontier S)"
31490
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31489
diff changeset
   891
  by (simp add: frontier_def closed_Diff)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   892
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   893
lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(UNIV - S))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   894
  by (auto simp add: frontier_def interior_closure)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   895
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   896
lemma frontier_straddle:
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   897
  fixes a :: "'a::metric_space"
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   898
  shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))" (is "?lhs \<longleftrightarrow> ?rhs")
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   899
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   900
  assume "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   901
  { fix e::real
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   902
    assume "e > 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   903
    let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   904
    { assume "a\<in>S"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   905
      have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   906
      moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   907
	unfolding frontier_closures closure_def islimpt_def using `e>0`
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   908
	by (auto, erule_tac x="ball a e" in allE, auto)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   909
      ultimately have ?rhse by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   910
    }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   911
    moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   912
    { assume "a\<notin>S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   913
      hence ?rhse using `?lhs`
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   914
	unfolding frontier_closures closure_def islimpt_def
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   915
	using open_ball[of a e] `e > 0`
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   916
	by (auto, erule_tac x = "ball a e" in allE, auto) (* FIXME: VERY slow! *)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   917
    }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   918
    ultimately have ?rhse by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   919
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   920
  thus ?rhs by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   921
next
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   922
  assume ?rhs
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   923
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   924
  { fix T assume "a\<notin>S" and
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   925
    as:"\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)" "a \<notin> S" "a \<in> T" "open T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   926
    from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   927
    then obtain e where "e>0" "ball a e \<subseteq> T" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   928
    then obtain y where y:"y\<in>S" "dist a y < e"  using as(1) by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   929
    have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   930
      using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   931
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   932
  hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   933
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   934
  { fix T assume "a \<in> T"  "open T" "a\<in>S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   935
    then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   936
    obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   937
    hence "\<exists>y\<in>UNIV - S. y \<in> T \<and> y \<noteq> a" using balle `a\<in>S` unfolding ball_def by (rule_tac x=x in bexI)auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   938
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   939
  hence "a islimpt (UNIV - S) \<or> a\<notin>S" unfolding islimpt_def by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   940
  ultimately show ?lhs unfolding frontier_closures using closure_def[of "UNIV - S"] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   941
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   942
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   943
lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   944
  by (metis frontier_def closure_closed Diff_subset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   945
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   946
lemma frontier_empty: "frontier {} = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   947
  by (simp add: frontier_def closure_empty)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   948
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   949
lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   950
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   951
  { assume "frontier S \<subseteq> S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   952
    hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   953
    hence "closed S" using closure_subset_eq by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   954
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   955
  thus ?thesis using frontier_subset_closed[of S] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   956
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   957
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   958
lemma frontier_complement: "frontier(UNIV - S) = frontier S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   959
  by (auto simp add: frontier_def closure_complement interior_complement)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   960
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   961
lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   962
  using frontier_complement frontier_subset_eq[of "UNIV - S"]
31490
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31489
diff changeset
   963
  unfolding open_closed Compl_eq_Diff_UNIV by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   964
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   965
subsection{* Common nets and The "within" modifier for nets. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   966
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   967
definition
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
   968
  at_infinity :: "'a::real_normed_vector net" where
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   969
  "at_infinity = Abs_net (range (\<lambda>r. {x. r \<le> norm x}))"
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
   970
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
   971
definition
31530
e31d63c66f55 generalize constant 'indirection'
huffman
parents: 31529
diff changeset
   972
  indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a net" (infixr "indirection" 70) where
e31d63c66f55 generalize constant 'indirection'
huffman
parents: 31529
diff changeset
   973
  "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   974
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   975
text{* Prove That They are all nets. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   976
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   977
lemma Rep_net_at_infinity:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   978
  "Rep_net at_infinity = range (\<lambda>r. {x. r \<le> norm x})"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   979
unfolding at_infinity_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   980
apply (rule Abs_net_inverse')
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   981
apply (rule image_nonempty, simp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   982
apply (clarsimp, rename_tac r s)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   983
apply (rule_tac x="max r s" in exI, auto)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   984
done
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   985
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
   986
lemma within_UNIV: "net within UNIV = net"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   987
  by (simp add: Rep_net_inject [symmetric] Rep_net_within)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   988
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   989
subsection{* Identify Trivial limits, where we can't approach arbitrarily closely. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   990
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
   991
definition
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
   992
  trivial_limit :: "'a net \<Rightarrow> bool" where
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   993
  "trivial_limit net \<longleftrightarrow> {} \<in> Rep_net net"
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
   994
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
   995
lemma trivial_limit_within:
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
   996
  shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   997
proof
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   998
  assume "trivial_limit (at a within S)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
   999
  thus "\<not> a islimpt S"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1000
    unfolding trivial_limit_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1001
    unfolding Rep_net_within Rep_net_at
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
  1002
    unfolding islimpt_def
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1003
    apply (clarsimp simp add: expand_set_eq)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1004
    apply (rename_tac T, rule_tac x=T in exI)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1005
    apply (clarsimp, drule_tac x=y in spec, simp)
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1006
    done
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1007
next
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1008
  assume "\<not> a islimpt S"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1009
  thus "trivial_limit (at a within S)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1010
    unfolding trivial_limit_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1011
    unfolding Rep_net_within Rep_net_at
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
  1012
    unfolding islimpt_def
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1013
    apply (clarsimp simp add: image_image)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1014
    apply (rule_tac x=T in image_eqI)
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1015
    apply (auto simp add: expand_set_eq)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1016
    done
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1017
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1018
31391
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1019
lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1020
  using trivial_limit_within [of a UNIV]
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1021
  by (simp add: within_UNIV)
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1022
31391
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1023
lemma trivial_limit_at:
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1024
  fixes a :: "'a::perfect_space"
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1025
  shows "\<not> trivial_limit (at a)"
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1026
  by (simp add: trivial_limit_at_iff)
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1027
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1028
lemma trivial_limit_at_infinity:
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1029
  "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,zero_neq_one}) net)"
31391
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1030
  (* FIXME: find a more appropriate type class *)
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1031
  unfolding trivial_limit_def Rep_net_at_infinity
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1032
  apply (clarsimp simp add: expand_set_eq)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1033
  apply (drule_tac x="scaleR r (sgn 1)" in spec)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1034
  apply (simp add: norm_scaleR norm_sgn)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1035
  done
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1036
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1037
lemma trivial_limit_sequentially: "\<not> trivial_limit sequentially"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1038
  by (auto simp add: trivial_limit_def Rep_net_sequentially)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1039
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1040
subsection{* Some property holds "sufficiently close" to the limit point. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1041
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1042
lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1043
  "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1044
unfolding eventually_at dist_nz by auto
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1045
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1046
lemma eventually_at_infinity:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1047
  "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1048
unfolding eventually_def Rep_net_at_infinity by auto
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1049
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1050
lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1051
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
31393
b8570dead501 reuse definition of nets from Limits.thy
huffman
parents: 31391
diff changeset
  1052
unfolding eventually_within eventually_at dist_nz by auto
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1053
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1054
lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1055
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1056
unfolding eventually_within
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1057
apply safe
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1058
apply (rule_tac x="d/2" in exI, simp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1059
apply (rule_tac x="d" in exI, simp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1060
done
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1061
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1062
lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1063
  unfolding eventually_def trivial_limit_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1064
  using Rep_net_nonempty [of net] by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1065
31347
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1066
lemma always_eventually: "(\<forall>x. P x) ==> eventually P net"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1067
  unfolding eventually_def trivial_limit_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1068
  using Rep_net_nonempty [of net] by auto
31347
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1069
31348
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1070
lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1071
  unfolding trivial_limit_def eventually_def by auto
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1072
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1073
lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1074
  unfolding trivial_limit_def eventually_def by auto
31348
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1075
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1076
lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1077
  apply (safe elim!: trivial_limit_eventually)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1078
  apply (simp add: eventually_False [symmetric])
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1079
  done
31348
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1080
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1081
text{* Combining theorems for "eventually" *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1082
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1083
lemma eventually_conjI:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1084
  "\<lbrakk>eventually (\<lambda>x. P x) net; eventually (\<lambda>x. Q x) net\<rbrakk>
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1085
    \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) net"
31393
b8570dead501 reuse definition of nets from Limits.thy
huffman
parents: 31391
diff changeset
  1086
by (rule eventually_conj)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1087
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1088
lemma eventually_rev_mono:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1089
  "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1090
using eventually_mono [of P Q] by fast
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1091
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1092
lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1093
  by (auto intro!: eventually_conjI elim: eventually_rev_mono)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1094
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1095
lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1096
  by (auto simp add: eventually_False)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1097
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1098
lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1099
  by (simp add: eventually_False)
31347
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1100
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1101
subsection{* Limits, defined as vacuously true when the limit is trivial. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1102
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1103
  text{* Notation Lim to avoid collition with lim defined in analysis *}
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
  1104
definition
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
  1105
  Lim :: "'a net \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'b" where
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
  1106
  "Lim net f = (THE l. (f ---> l) net)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1107
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1108
lemma Lim:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1109
 "(f ---> l) net \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1110
        trivial_limit net \<or>
31348
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1111
        (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
  1112
  unfolding tendsto_iff trivial_limit_eq by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1113
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1114
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1115
text{* Show that they yield usual definitions in the various cases. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1116
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1117
lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1118
           (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a  \<and> dist x a  <= d \<longrightarrow> dist (f x) l < e)"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
  1119
  by (auto simp add: tendsto_iff eventually_within_le)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1120
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1121
lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1122
        (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
  1123
  by (auto simp add: tendsto_iff eventually_within)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1124
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1125
lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1126
        (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
  1127
  by (auto simp add: tendsto_iff eventually_at)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1128
31342
b7941738e3a1 add lemmas Lim_at_iff_LIM, Lim_sequentially_iff_LIMSEQ
huffman
parents: 31341
diff changeset
  1129
lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
b7941738e3a1 add lemmas Lim_at_iff_LIM, Lim_sequentially_iff_LIMSEQ
huffman
parents: 31341
diff changeset
  1130
  unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
b7941738e3a1 add lemmas Lim_at_iff_LIM, Lim_sequentially_iff_LIMSEQ
huffman
parents: 31341
diff changeset
  1131
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1132
lemma Lim_at_infinity:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1133
  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x::real^'n::finite. norm x >= b \<longrightarrow> dist (f x) l < e)"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
  1134
  by (auto simp add: tendsto_iff eventually_at_infinity)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1135
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1136
lemma Lim_sequentially:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1137
 "(S ---> l) sequentially \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1138
          (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
  1139
  by (auto simp add: tendsto_iff eventually_sequentially)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1140
31342
b7941738e3a1 add lemmas Lim_at_iff_LIM, Lim_sequentially_iff_LIMSEQ
huffman
parents: 31341
diff changeset
  1141
lemma Lim_sequentially_iff_LIMSEQ: "(S ---> l) sequentially \<longleftrightarrow> S ----> l"
b7941738e3a1 add lemmas Lim_at_iff_LIM, Lim_sequentially_iff_LIMSEQ
huffman
parents: 31341
diff changeset
  1142
  unfolding Lim_sequentially LIMSEQ_def ..
b7941738e3a1 add lemmas Lim_at_iff_LIM, Lim_sequentially_iff_LIMSEQ
huffman
parents: 31341
diff changeset
  1143
31525
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1144
lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1145
  by (rule topological_tendstoI, auto elim: eventually_rev_mono)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1146
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1147
text{* The expected monotonicity property. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1148
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1149
lemma Lim_within_empty: "(f ---> l) (net within {})"
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1150
  unfolding tendsto_def Limits.eventually_within by simp
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1151
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1152
lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1153
  unfolding tendsto_def Limits.eventually_within
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1154
  by (auto elim!: eventually_elim1)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1155
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1156
lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1157
  shows "(f ---> l) (net within (S \<union> T))"
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1158
  using assms unfolding tendsto_def Limits.eventually_within
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1159
  apply clarify
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
  1160
  apply (drule spec, drule (1) mp, drule (1) mp)
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
  1161
  apply (drule spec, drule (1) mp, drule (1) mp)
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1162
  apply (auto elim: eventually_elim2)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1163
  done
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1164
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1165
lemma Lim_Un_univ:
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1166
 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1167
        ==> (f ---> l) net"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1168
  by (metis Lim_Un within_UNIV)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1169
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1170
text{* Interrelations between restricted and unrestricted limits. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1171
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1172
lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
31525
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1173
  (* FIXME: rename *)
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1174
  unfolding tendsto_def Limits.eventually_within
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
  1175
  apply (clarify, drule spec, drule (1) mp, drule (1) mp)
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1176
  by (auto elim!: eventually_elim1)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1177
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1178
lemma Lim_within_open:
31525
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1179
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1180
  assumes"a \<in> S" "open S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1181
  shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1182
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1183
  assume ?lhs
31525
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1184
  { fix A assume "open A" "l \<in> A"
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1185
    with `?lhs` have "eventually (\<lambda>x. f x \<in> A) (at a within S)"
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1186
      by (rule topological_tendstoD)
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1187
    hence "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x \<in> A) (at a)"
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1188
      unfolding Limits.eventually_within .
31525
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1189
    then obtain T where "open T" "a \<in> T" "\<forall>x\<in>T. x \<noteq> a \<longrightarrow> x \<in> S \<longrightarrow> f x \<in> A"
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
  1190
      unfolding eventually_at_topological by fast
31525
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1191
    hence "open (T \<inter> S)" "a \<in> T \<inter> S" "\<forall>x\<in>(T \<inter> S). x \<noteq> a \<longrightarrow> f x \<in> A"
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1192
      using assms by auto
31525
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1193
    hence "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> f x \<in> A)"
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1194
      by fast
31525
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1195
    hence "eventually (\<lambda>x. f x \<in> A) (at a)"
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
  1196
      unfolding eventually_at_topological .
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1197
  }
31525
472b844f8607 generalize some lemmas
huffman
parents: 31492
diff changeset
  1198
  thus ?rhs by (rule topological_tendstoI)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1199
next
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1200
  assume ?rhs
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1201
  thus ?lhs by (rule Lim_at_within)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1202
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1203
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1204
text{* Another limit point characterization. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1205
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1206
lemma islimpt_sequential:
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
  1207
  fixes x :: "'a::metric_space" (* FIXME: generalize to topological_space *)
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
  1208
  shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
  1209
    (is "?lhs = ?rhs")
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1210
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1211
  assume ?lhs