src/HOL/Analysis/Elementary_Topology.thy
 author wenzelm Mon Mar 25 17:21:26 2019 +0100 (4 weeks ago) changeset 69981 3dced198b9ec parent 69768 7e4966eaf781 child 70044 da5857dbcbb9 permissions -rw-r--r--
more strict AFP properties;
1 (*  Author:     L C Paulson, University of Cambridge
2     Author:     Amine Chaieb, University of Cambridge
3     Author:     Robert Himmelmann, TU Muenchen
4     Author:     Brian Huffman, Portland State University
5 *)
7 chapter \<open>Topology\<close>
9 theory Elementary_Topology
10 imports
11   "HOL-Library.Set_Idioms"
12   "HOL-Library.Disjoint_Sets"
13   Product_Vector
14 begin
17 section \<open>Elementary Topology\<close>
19 subsection \<open>TODO: move?\<close>
21 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
22   using openI by auto
25 subsubsection%unimportant \<open>Archimedean properties and useful consequences\<close>
27 text\<open>Bernoulli's inequality\<close>
28 proposition Bernoulli_inequality:
29   fixes x :: real
30   assumes "-1 \<le> x"
31     shows "1 + n * x \<le> (1 + x) ^ n"
32 proof (induct n)
33   case 0
34   then show ?case by simp
35 next
36   case (Suc n)
37   have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
39   also have "... = (1 + x) * (1 + n*x)"
40     by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
41   also have "... \<le> (1 + x) ^ Suc n"
42     using Suc.hyps assms mult_left_mono by fastforce
43   finally show ?case .
44 qed
46 corollary Bernoulli_inequality_even:
47   fixes x :: real
48   assumes "even n"
49     shows "1 + n * x \<le> (1 + x) ^ n"
50 proof (cases "-1 \<le> x \<or> n=0")
51   case True
52   then show ?thesis
53     by (auto simp: Bernoulli_inequality)
54 next
55   case False
56   then have "real n \<ge> 1"
57     by simp
58   with False have "n * x \<le> -1"
59     by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
60   then have "1 + n * x \<le> 0"
61     by auto
62   also have "... \<le> (1 + x) ^ n"
63     using assms
64     using zero_le_even_power by blast
65   finally show ?thesis .
66 qed
68 corollary real_arch_pow:
69   fixes x :: real
70   assumes x: "1 < x"
71   shows "\<exists>n. y < x^n"
72 proof -
73   from x have x0: "x - 1 > 0"
74     by arith
75   from reals_Archimedean3[OF x0, rule_format, of y]
76   obtain n :: nat where n: "y < real n * (x - 1)" by metis
77   from x0 have x00: "x- 1 \<ge> -1" by arith
78   from Bernoulli_inequality[OF x00, of n] n
79   have "y < x^n" by auto
80   then show ?thesis by metis
81 qed
83 corollary real_arch_pow_inv:
84   fixes x y :: real
85   assumes y: "y > 0"
86     and x1: "x < 1"
87   shows "\<exists>n. x^n < y"
88 proof (cases "x > 0")
89   case True
90   with x1 have ix: "1 < 1/x" by (simp add: field_simps)
91   from real_arch_pow[OF ix, of "1/y"]
92   obtain n where n: "1/y < (1/x)^n" by blast
93   then show ?thesis using y \<open>x > 0\<close>
94     by (auto simp add: field_simps)
95 next
96   case False
97   with y x1 show ?thesis
98     by (metis less_le_trans not_less power_one_right)
99 qed
101 lemma forall_pos_mono:
102   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
103     (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
104   by (metis real_arch_inverse)
106 lemma forall_pos_mono_1:
107   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
108     (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
109   apply (rule forall_pos_mono)
110   apply auto
111   apply (metis Suc_pred of_nat_Suc)
112   done
114 subsubsection%unimportant \<open>Affine transformations of intervals\<close>
116 lemma real_affinity_le: "0 < m \<Longrightarrow> m * x + c \<le> y \<longleftrightarrow> x \<le> inverse m * y + - (c / m)"
117   for m :: "'a::linordered_field"
120 lemma real_le_affinity: "0 < m \<Longrightarrow> y \<le> m * x + c \<longleftrightarrow> inverse m * y + - (c / m) \<le> x"
121   for m :: "'a::linordered_field"
124 lemma real_affinity_lt: "0 < m \<Longrightarrow> m * x + c < y \<longleftrightarrow> x < inverse m * y + - (c / m)"
125   for m :: "'a::linordered_field"
128 lemma real_lt_affinity: "0 < m \<Longrightarrow> y < m * x + c \<longleftrightarrow> inverse m * y + - (c / m) < x"
129   for m :: "'a::linordered_field"
132 lemma real_affinity_eq: "m \<noteq> 0 \<Longrightarrow> m * x + c = y \<longleftrightarrow> x = inverse m * y + - (c / m)"
133   for m :: "'a::linordered_field"
136 lemma real_eq_affinity: "m \<noteq> 0 \<Longrightarrow> y = m * x + c  \<longleftrightarrow> inverse m * y + - (c / m) = x"
137   for m :: "'a::linordered_field"
142 subsection \<open>Topological Basis\<close>
144 context topological_space
145 begin
147 definition%important "topological_basis B \<longleftrightarrow>
148   (\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
150 lemma topological_basis:
151   "topological_basis B \<longleftrightarrow> (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
152   unfolding topological_basis_def
153   apply safe
154      apply fastforce
155     apply fastforce
156    apply (erule_tac x=x in allE, simp)
157    apply (rule_tac x="{x}" in exI, auto)
158   done
160 lemma topological_basis_iff:
161   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
162   shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
163     (is "_ \<longleftrightarrow> ?rhs")
164 proof safe
165   fix O' and x::'a
166   assume H: "topological_basis B" "open O'" "x \<in> O'"
167   then have "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
168   then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto
169   then show "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
170 next
171   assume H: ?rhs
172   show "topological_basis B"
173     using assms unfolding topological_basis_def
174   proof safe
175     fix O' :: "'a set"
176     assume "open O'"
177     with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
178       by (force intro: bchoice simp: Bex_def)
179     then show "\<exists>B'\<subseteq>B. \<Union>B' = O'"
180       by (auto intro: exI[where x="{f x |x. x \<in> O'}"])
181   qed
182 qed
184 lemma topological_basisI:
185   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
186     and "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
187   shows "topological_basis B"
188   using assms by (subst topological_basis_iff) auto
190 lemma topological_basisE:
191   fixes O'
192   assumes "topological_basis B"
193     and "open O'"
194     and "x \<in> O'"
195   obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"
196 proof atomize_elim
197   from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'"
199   with topological_basis_iff assms
200   show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'"
201     using assms by (simp add: Bex_def)
202 qed
204 lemma topological_basis_open:
205   assumes "topological_basis B"
206     and "X \<in> B"
207   shows "open X"
208   using assms by (simp add: topological_basis_def)
210 lemma topological_basis_imp_subbasis:
211   assumes B: "topological_basis B"
212   shows "open = generate_topology B"
213 proof (intro ext iffI)
214   fix S :: "'a set"
215   assume "open S"
216   with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"
217     unfolding topological_basis_def by blast
218   then show "generate_topology B S"
219     by (auto intro: generate_topology.intros dest: topological_basis_open)
220 next
221   fix S :: "'a set"
222   assume "generate_topology B S"
223   then show "open S"
224     by induct (auto dest: topological_basis_open[OF B])
225 qed
227 lemma basis_dense:
228   fixes B :: "'a set set"
229     and f :: "'a set \<Rightarrow> 'a"
230   assumes "topological_basis B"
231     and choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
232   shows "\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X)"
233 proof (intro allI impI)
234   fix X :: "'a set"
235   assume "open X" and "X \<noteq> {}"
236   from topological_basisE[OF \<open>topological_basis B\<close> \<open>open X\<close> choosefrom_basis[OF \<open>X \<noteq> {}\<close>]]
237   obtain B' where "B' \<in> B" "f X \<in> B'" "B' \<subseteq> X" .
238   then show "\<exists>B'\<in>B. f B' \<in> X"
239     by (auto intro!: choosefrom_basis)
240 qed
242 end
244 lemma topological_basis_prod:
245   assumes A: "topological_basis A"
246     and B: "topological_basis B"
247   shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
248   unfolding topological_basis_def
249 proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
250   fix S :: "('a \<times> 'b) set"
251   assume "open S"
252   then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
253   proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
254     fix x y
255     assume "(x, y) \<in> S"
256     from open_prod_elim[OF \<open>open S\<close> this]
257     obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
258       by (metis mem_Sigma_iff)
259     moreover
260     from A a obtain A0 where "A0 \<in> A" "x \<in> A0" "A0 \<subseteq> a"
261       by (rule topological_basisE)
262     moreover
263     from B b obtain B0 where "B0 \<in> B" "y \<in> B0" "B0 \<subseteq> b"
264       by (rule topological_basisE)
265     ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
266       by (intro UN_I[of "(A0, B0)"]) auto
267   qed auto
268 qed (metis A B topological_basis_open open_Times)
271 subsection \<open>Countable Basis\<close>
273 locale%important countable_basis = topological_space p for p::"'a set \<Rightarrow> bool" +
274   fixes B :: "'a set set"
275   assumes is_basis: "topological_basis B"
276     and countable_basis: "countable B"
277 begin
279 lemma open_countable_basis_ex:
280   assumes "p X"
281   shows "\<exists>B' \<subseteq> B. X = \<Union>B'"
282   using assms countable_basis is_basis
283   unfolding topological_basis_def by blast
285 lemma open_countable_basisE:
286   assumes "p X"
287   obtains B' where "B' \<subseteq> B" "X = \<Union>B'"
288   using assms open_countable_basis_ex
289   by atomize_elim simp
291 lemma countable_dense_exists:
292   "\<exists>D::'a set. countable D \<and> (\<forall>X. p X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"
293 proof -
294   let ?f = "(\<lambda>B'. SOME x. x \<in> B')"
295   have "countable (?f ` B)" using countable_basis by simp
296   with basis_dense[OF is_basis, of ?f] show ?thesis
297     by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)
298 qed
300 lemma countable_dense_setE:
301   obtains D :: "'a set"
302   where "countable D" "\<And>X. p X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"
303   using countable_dense_exists by blast
305 end
307 lemma countable_basis_openI: "countable_basis open B"
308   if "countable B" "topological_basis B"
309   using that
310   by unfold_locales
311     (simp_all add: topological_basis topological_space.topological_basis topological_space_axioms)
313 lemma (in first_countable_topology) first_countable_basisE:
314   fixes x :: 'a
315   obtains \<A> where "countable \<A>" "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
316     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)"
317 proof -
318   obtain \<A> where \<A>: "(\<forall>i::nat. x \<in> \<A> i \<and> open (\<A> i))" "(\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
319     using first_countable_basis[of x] by metis
320   show thesis
321   proof
322     show "countable (range \<A>)"
323       by simp
324   qed (use \<A> in auto)
325 qed
327 lemma (in first_countable_topology) first_countable_basis_Int_stableE:
328   obtains \<A> where "countable \<A>" "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
329     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)"
330     "\<And>A B. A \<in> \<A> \<Longrightarrow> B \<in> \<A> \<Longrightarrow> A \<inter> B \<in> \<A>"
331 proof atomize_elim
332   obtain \<B> where \<B>:
333     "countable \<B>"
334     "\<And>B. B \<in> \<B> \<Longrightarrow> x \<in> B"
335     "\<And>B. B \<in> \<B> \<Longrightarrow> open B"
336     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>B\<in>\<B>. B \<subseteq> S"
337     by (rule first_countable_basisE) blast
338   define \<A> where [abs_def]:
339     "\<A> = (\<lambda>N. \<Inter>((\<lambda>n. from_nat_into \<B> n) ` N)) ` (Collect finite::nat set set)"
340   then show "\<exists>\<A>. countable \<A> \<and> (\<forall>A. A \<in> \<A> \<longrightarrow> x \<in> A) \<and> (\<forall>A. A \<in> \<A> \<longrightarrow> open A) \<and>
341         (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)) \<and> (\<forall>A B. A \<in> \<A> \<longrightarrow> B \<in> \<A> \<longrightarrow> A \<inter> B \<in> \<A>)"
342   proof (safe intro!: exI[where x=\<A>])
343     show "countable \<A>"
344       unfolding \<A>_def by (intro countable_image countable_Collect_finite)
345     fix A
346     assume "A \<in> \<A>"
347     then show "x \<in> A" "open A"
348       using \<B>(4)[OF open_UNIV] by (auto simp: \<A>_def intro: \<B> from_nat_into)
349   next
350     let ?int = "\<lambda>N. \<Inter>(from_nat_into \<B> ` N)"
351     fix A B
352     assume "A \<in> \<A>" "B \<in> \<A>"
353     then obtain N M where "A = ?int N" "B = ?int M" "finite (N \<union> M)"
354       by (auto simp: \<A>_def)
355     then show "A \<inter> B \<in> \<A>"
356       by (auto simp: \<A>_def intro!: image_eqI[where x="N \<union> M"])
357   next
358     fix S
359     assume "open S" "x \<in> S"
360     then obtain a where a: "a\<in>\<B>" "a \<subseteq> S" using \<B> by blast
361     then show "\<exists>a\<in>\<A>. a \<subseteq> S" using a \<B>
362       by (intro bexI[where x=a]) (auto simp: \<A>_def intro: image_eqI[where x="{to_nat_on \<B> a}"])
363   qed
364 qed
366 lemma (in topological_space) first_countableI:
367   assumes "countable \<A>"
368     and 1: "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
369     and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>A\<in>\<A>. A \<subseteq> S"
370   shows "\<exists>\<A>::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> \<A> i \<and> open (\<A> i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
371 proof (safe intro!: exI[of _ "from_nat_into \<A>"])
372   fix i
373   have "\<A> \<noteq> {}" using 2[of UNIV] by auto
374   show "x \<in> from_nat_into \<A> i" "open (from_nat_into \<A> i)"
375     using range_from_nat_into_subset[OF \<open>\<A> \<noteq> {}\<close>] 1 by auto
376 next
377   fix S
378   assume "open S" "x\<in>S" from 2[OF this]
379   show "\<exists>i. from_nat_into \<A> i \<subseteq> S"
380     using subset_range_from_nat_into[OF \<open>countable \<A>\<close>] by auto
381 qed
383 instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
384 proof
385   fix x :: "'a \<times> 'b"
386   obtain \<A> where \<A>:
387       "countable \<A>"
388       "\<And>a. a \<in> \<A> \<Longrightarrow> fst x \<in> a"
389       "\<And>a. a \<in> \<A> \<Longrightarrow> open a"
390       "\<And>S. open S \<Longrightarrow> fst x \<in> S \<Longrightarrow> \<exists>a\<in>\<A>. a \<subseteq> S"
391     by (rule first_countable_basisE[of "fst x"]) blast
392   obtain B where B:
393       "countable B"
394       "\<And>a. a \<in> B \<Longrightarrow> snd x \<in> a"
395       "\<And>a. a \<in> B \<Longrightarrow> open a"
396       "\<And>S. open S \<Longrightarrow> snd x \<in> S \<Longrightarrow> \<exists>a\<in>B. a \<subseteq> S"
397     by (rule first_countable_basisE[of "snd x"]) blast
398   show "\<exists>\<A>::nat \<Rightarrow> ('a \<times> 'b) set.
399     (\<forall>i. x \<in> \<A> i \<and> open (\<A> i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
400   proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (\<A> \<times> B)"], safe)
401     fix a b
402     assume x: "a \<in> \<A>" "b \<in> B"
403     show "x \<in> a \<times> b"
404       by (simp add: \<A>(2) B(2) mem_Times_iff x)
405     show "open (a \<times> b)"
406       by (simp add: \<A>(3) B(3) open_Times x)
407   next
408     fix S
409     assume "open S" "x \<in> S"
410     then obtain a' b' where a'b': "open a'" "open b'" "x \<in> a' \<times> b'" "a' \<times> b' \<subseteq> S"
411       by (rule open_prod_elim)
412     moreover
413     from a'b' \<A>(4)[of a'] B(4)[of b']
414     obtain a b where "a \<in> \<A>" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'"
415       by auto
416     ultimately
417     show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (\<A> \<times> B). a \<subseteq> S"
418       by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
419   qed (simp add: \<A> B)
420 qed
422 class second_countable_topology = topological_space +
423   assumes ex_countable_subbasis:
424     "\<exists>B::'a set set. countable B \<and> open = generate_topology B"
425 begin
427 lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
428 proof -
429   from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"
430     by blast
431   let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
433   show ?thesis
434   proof (intro exI conjI)
435     show "countable ?B"
436       by (intro countable_image countable_Collect_finite_subset B)
437     {
438       fix S
439       assume "open S"
440       then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
441         unfolding B
442       proof induct
443         case UNIV
444         show ?case by (intro exI[of _ "{{}}"]) simp
445       next
446         case (Int a b)
447         then obtain x y where x: "a = \<Union>(Inter ` x)" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
448           and y: "b = \<Union>(Inter ` y)" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
449           by blast
450         show ?case
451           unfolding x y Int_UN_distrib2
452           by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
453       next
454         case (UN K)
455         then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. \<Union> (Inter ` B') = k" by auto
456         then obtain k where
457             "\<forall>ka\<in>K. k ka \<subseteq> {b. finite b \<and> b \<subseteq> B} \<and> \<Union>(Inter ` (k ka)) = ka"
458           unfolding bchoice_iff ..
459         then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. \<Union> (Inter ` B') = \<Union>K"
460           by (intro exI[of _ "\<Union>(k ` K)"]) auto
461       next
462         case (Basis S)
463         then show ?case
464           by (intro exI[of _ "{{S}}"]) auto
465       qed
466       then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
467         unfolding subset_image_iff by blast }
468     then show "topological_basis ?B"
469       unfolding topological_basis_def
470       by (safe intro!: open_Inter)
471          (simp_all add: B generate_topology.Basis subset_eq)
472   qed
473 qed
476 end
478 lemma univ_second_countable:
479   obtains \<B> :: "'a::second_countable_topology set set"
480   where "countable \<B>" "\<And>C. C \<in> \<B> \<Longrightarrow> open C"
481        "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"
482 by (metis ex_countable_basis topological_basis_def)
484 proposition Lindelof:
485   fixes \<F> :: "'a::second_countable_topology set set"
486   assumes \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> open S"
487   obtains \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
488 proof -
489   obtain \<B> :: "'a set set"
490     where "countable \<B>" "\<And>C. C \<in> \<B> \<Longrightarrow> open C"
491       and \<B>: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"
492     using univ_second_countable by blast
493   define \<D> where "\<D> \<equiv> {S. S \<in> \<B> \<and> (\<exists>U. U \<in> \<F> \<and> S \<subseteq> U)}"
494   have "countable \<D>"
495     apply (rule countable_subset [OF _ \<open>countable \<B>\<close>])
496     apply (force simp: \<D>_def)
497     done
498   have "\<And>S. \<exists>U. S \<in> \<D> \<longrightarrow> U \<in> \<F> \<and> S \<subseteq> U"
500   then obtain G where G: "\<And>S. S \<in> \<D> \<longrightarrow> G S \<in> \<F> \<and> S \<subseteq> G S"
501     by metis
502   have "\<Union>\<F> \<subseteq> \<Union>\<D>"
503     unfolding \<D>_def by (blast dest: \<F> \<B>)
504   moreover have "\<Union>\<D> \<subseteq> \<Union>\<F>"
505     using \<D>_def by blast
506   ultimately have eq1: "\<Union>\<F> = \<Union>\<D>" ..
507   have eq2: "\<Union>\<D> = \<Union> (G ` \<D>)"
508     using G eq1 by auto
509   show ?thesis
510     apply (rule_tac \<F>' = "G ` \<D>" in that)
511     using G \<open>countable \<D>\<close>
512     by (auto simp: eq1 eq2)
513 qed
515 lemma countable_disjoint_open_subsets:
516   fixes \<F> :: "'a::second_countable_topology set set"
517   assumes "\<And>S. S \<in> \<F> \<Longrightarrow> open S" and pw: "pairwise disjnt \<F>"
518     shows "countable \<F>"
519 proof -
520   obtain \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
521     by (meson assms Lindelof)
522   with pw have "\<F> \<subseteq> insert {} \<F>'"
523     by (fastforce simp add: pairwise_def disjnt_iff)
524   then show ?thesis
525     by (simp add: \<open>countable \<F>'\<close> countable_subset)
526 qed
528 sublocale second_countable_topology <
529   countable_basis "open" "SOME B. countable B \<and> topological_basis B"
530   using someI_ex[OF ex_countable_basis]
531   by unfold_locales safe
534 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
535 proof
536   obtain A :: "'a set set" where "countable A" "topological_basis A"
537     using ex_countable_basis by auto
538   moreover
539   obtain B :: "'b set set" where "countable B" "topological_basis B"
540     using ex_countable_basis by auto
541   ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
542     by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
543       topological_basis_imp_subbasis)
544 qed
546 instance second_countable_topology \<subseteq> first_countable_topology
547 proof
548   fix x :: 'a
549   define B :: "'a set set" where "B = (SOME B. countable B \<and> topological_basis B)"
550   then have B: "countable B" "topological_basis B"
551     using countable_basis is_basis
552     by (auto simp: countable_basis is_basis)
553   then show "\<exists>A::nat \<Rightarrow> 'a set.
554     (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
555     by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
556        (fastforce simp: topological_space_class.topological_basis_def)+
557 qed
559 instance nat :: second_countable_topology
560 proof
561   show "\<exists>B::nat set set. countable B \<and> open = generate_topology B"
562     by (intro exI[of _ "range lessThan \<union> range greaterThan"]) (auto simp: open_nat_def)
563 qed
565 lemma countable_separating_set_linorder1:
566   shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y))"
567 proof -
568   obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
569   define B1 where "B1 = {(LEAST x. x \<in> U)| U. U \<in> A}"
570   then have "countable B1" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
571   define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
572   then have "countable B2" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
573   have "\<exists>b \<in> B1 \<union> B2. x < b \<and> b \<le> y" if "x < y" for x y
574   proof (cases)
575     assume "\<exists>z. x < z \<and> z < y"
576     then obtain z where z: "x < z \<and> z < y" by auto
577     define U where "U = {x<..<y}"
578     then have "open U" by simp
579     moreover have "z \<in> U" using z U_def by simp
580     ultimately obtain V where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
581     define w where "w = (SOME x. x \<in> V)"
582     then have "w \<in> V" using \<open>z \<in> V\<close> by (metis someI2)
583     then have "x < w \<and> w \<le> y" using \<open>w \<in> V\<close> \<open>V \<subseteq> U\<close> U_def by fastforce
584     moreover have "w \<in> B1 \<union> B2" using w_def B2_def \<open>V \<in> A\<close> by auto
585     ultimately show ?thesis by auto
586   next
587     assume "\<not>(\<exists>z. x < z \<and> z < y)"
588     then have *: "\<And>z. z > x \<Longrightarrow> z \<ge> y" by auto
589     define U where "U = {x<..}"
590     then have "open U" by simp
591     moreover have "y \<in> U" using \<open>x < y\<close> U_def by simp
592     ultimately obtain "V" where "V \<in> A" "y \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
593     have "U = {y..}" unfolding U_def using * \<open>x < y\<close> by auto
594     then have "V \<subseteq> {y..}" using \<open>V \<subseteq> U\<close> by simp
595     then have "(LEAST w. w \<in> V) = y" using \<open>y \<in> V\<close> by (meson Least_equality atLeast_iff subsetCE)
596     then have "y \<in> B1 \<union> B2" using \<open>V \<in> A\<close> B1_def by auto
597     moreover have "x < y \<and> y \<le> y" using \<open>x < y\<close> by simp
598     ultimately show ?thesis by auto
599   qed
600   moreover have "countable (B1 \<union> B2)" using \<open>countable B1\<close> \<open>countable B2\<close> by simp
601   ultimately show ?thesis by auto
602 qed
604 lemma countable_separating_set_linorder2:
605   shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x \<le> b \<and> b < y))"
606 proof -
607   obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
608   define B1 where "B1 = {(GREATEST x. x \<in> U) | U. U \<in> A}"
609   then have "countable B1" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
610   define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
611   then have "countable B2" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
612   have "\<exists>b \<in> B1 \<union> B2. x \<le> b \<and> b < y" if "x < y" for x y
613   proof (cases)
614     assume "\<exists>z. x < z \<and> z < y"
615     then obtain z where z: "x < z \<and> z < y" by auto
616     define U where "U = {x<..<y}"
617     then have "open U" by simp
618     moreover have "z \<in> U" using z U_def by simp
619     ultimately obtain "V" where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
620     define w where "w = (SOME x. x \<in> V)"
621     then have "w \<in> V" using \<open>z \<in> V\<close> by (metis someI2)
622     then have "x \<le> w \<and> w < y" using \<open>w \<in> V\<close> \<open>V \<subseteq> U\<close> U_def by fastforce
623     moreover have "w \<in> B1 \<union> B2" using w_def B2_def \<open>V \<in> A\<close> by auto
624     ultimately show ?thesis by auto
625   next
626     assume "\<not>(\<exists>z. x < z \<and> z < y)"
627     then have *: "\<And>z. z < y \<Longrightarrow> z \<le> x" using leI by blast
628     define U where "U = {..<y}"
629     then have "open U" by simp
630     moreover have "x \<in> U" using \<open>x < y\<close> U_def by simp
631     ultimately obtain "V" where "V \<in> A" "x \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
632     have "U = {..x}" unfolding U_def using * \<open>x < y\<close> by auto
633     then have "V \<subseteq> {..x}" using \<open>V \<subseteq> U\<close> by simp
634     then have "(GREATEST x. x \<in> V) = x" using \<open>x \<in> V\<close> by (meson Greatest_equality atMost_iff subsetCE)
635     then have "x \<in> B1 \<union> B2" using \<open>V \<in> A\<close> B1_def by auto
636     moreover have "x \<le> x \<and> x < y" using \<open>x < y\<close> by simp
637     ultimately show ?thesis by auto
638   qed
639   moreover have "countable (B1 \<union> B2)" using \<open>countable B1\<close> \<open>countable B2\<close> by simp
640   ultimately show ?thesis by auto
641 qed
643 lemma countable_separating_set_dense_linorder:
644   shows "\<exists>B::('a::{linorder_topology, dense_linorder, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b < y))"
645 proof -
646   obtain B::"'a set" where B: "countable B" "\<And>x y. x < y \<Longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y)"
647     using countable_separating_set_linorder1 by auto
648   have "\<exists>b \<in> B. x < b \<and> b < y" if "x < y" for x y
649   proof -
650     obtain z where "x < z" "z < y" using \<open>x < y\<close> dense by blast
651     then obtain b where "b \<in> B" "x < b \<and> b \<le> z" using B(2) by auto
652     then have "x < b \<and> b < y" using \<open>z < y\<close> by auto
653     then show ?thesis using \<open>b \<in> B\<close> by auto
654   qed
655   then show ?thesis using B(1) by auto
656 qed
659 subsection \<open>Polish spaces\<close>
661 text \<open>Textbooks define Polish spaces as completely metrizable.
662   We assume the topology to be complete for a given metric.\<close>
664 class polish_space = complete_space + second_countable_topology
667 subsection \<open>Limit Points\<close>
669 definition%important (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infixr "islimpt" 60)
670   where "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
672 lemma islimptI:
673   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
674   shows "x islimpt S"
675   using assms unfolding islimpt_def by auto
677 lemma islimptE:
678   assumes "x islimpt S" and "x \<in> T" and "open T"
679   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
680   using assms unfolding islimpt_def by auto
682 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
683   unfolding islimpt_def eventually_at_topological by auto
685 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"
686   unfolding islimpt_def by fast
688 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
689   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
691 lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
692   unfolding islimpt_def by blast
694 text \<open>A perfect space has no isolated points.\<close>
696 lemma islimpt_UNIV [simp, intro]: "x islimpt UNIV"
697   for x :: "'a::perfect_space"
698   unfolding islimpt_UNIV_iff by (rule not_open_singleton)
700 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
701   unfolding closed_def
702   apply (subst open_subopen)
703   apply (simp add: islimpt_def subset_eq)
704   apply (metis ComplE ComplI)
705   done
707 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
708   by (auto simp: islimpt_def)
710 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
711   by (simp add: islimpt_iff_eventually eventually_conj_iff)
714 lemma islimpt_insert:
715   fixes x :: "'a::t1_space"
716   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
717 proof
718   assume *: "x islimpt (insert a s)"
719   show "x islimpt s"
720   proof (rule islimptI)
721     fix t
722     assume t: "x \<in> t" "open t"
723     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
724     proof (cases "x = a")
725       case True
726       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
727         using * t by (rule islimptE)
728       with \<open>x = a\<close> show ?thesis by auto
729     next
730       case False
731       with t have t': "x \<in> t - {a}" "open (t - {a})"
733       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
734         using * t' by (rule islimptE)
735       then show ?thesis by auto
736     qed
737   qed
738 next
739   assume "x islimpt s"
740   then show "x islimpt (insert a s)"
741     by (rule islimpt_subset) auto
742 qed
744 lemma islimpt_finite:
745   fixes x :: "'a::t1_space"
746   shows "finite s \<Longrightarrow> \<not> x islimpt s"
747   by (induct set: finite) (simp_all add: islimpt_insert)
749 lemma islimpt_Un_finite:
750   fixes x :: "'a::t1_space"
751   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
752   by (simp add: islimpt_Un islimpt_finite)
754 lemma islimpt_eq_acc_point:
755   fixes l :: "'a :: t1_space"
756   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
757 proof (safe intro!: islimptI)
758   fix U
759   assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
760   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"
761     by (auto intro: finite_imp_closed)
762   then show False
763     by (rule islimptE) auto
764 next
765   fix T
766   assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
767   then have "infinite (T \<inter> S - {l})"
768     by auto
769   then have "\<exists>x. x \<in> (T \<inter> S - {l})"
770     unfolding ex_in_conv by (intro notI) simp
771   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"
772     by auto
773 qed
775 lemma acc_point_range_imp_convergent_subsequence:
776   fixes l :: "'a :: first_countable_topology"
777   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
778   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
779 proof -
780   from countable_basis_at_decseq[of l]
781   obtain A where A:
782       "\<And>i. open (A i)"
783       "\<And>i. l \<in> A i"
784       "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
785     by blast
786   define s where "s n i = (SOME j. i < j \<and> f j \<in> A (Suc n))" for n i
787   {
788     fix n i
789     have "infinite (A (Suc n) \<inter> range f - f`{.. i})"
790       using l A by auto
791     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f`{.. i}"
792       unfolding ex_in_conv by (intro notI) simp
793     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"
794       by auto
795     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"
796       by (auto simp: not_le)
797     then have "i < s n i" "f (s n i) \<in> A (Suc n)"
798       unfolding s_def by (auto intro: someI2_ex)
799   }
800   note s = this
801   define r where "r = rec_nat (s 0 0) s"
802   have "strict_mono r"
803     by (auto simp: r_def s strict_mono_Suc_iff)
804   moreover
805   have "(\<lambda>n. f (r n)) \<longlonglongrightarrow> l"
806   proof (rule topological_tendstoI)
807     fix S
808     assume "open S" "l \<in> S"
809     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
810       by auto
811     moreover
812     {
813       fix i
814       assume "Suc 0 \<le> i"
815       then have "f (r i) \<in> A i"
816         by (cases i) (simp_all add: r_def s)
817     }
818     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"
819       by (auto simp: eventually_sequentially)
820     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"
821       by eventually_elim auto
822   qed
823   ultimately show "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
824     by (auto simp: convergent_def comp_def)
825 qed
827 lemma islimpt_range_imp_convergent_subsequence:
828   fixes l :: "'a :: {t1_space, first_countable_topology}"
829   assumes l: "l islimpt (range f)"
830   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
831   using l unfolding islimpt_eq_acc_point
832   by (rule acc_point_range_imp_convergent_subsequence)
834 lemma sequence_unique_limpt:
835   fixes f :: "nat \<Rightarrow> 'a::t2_space"
836   assumes "(f \<longlongrightarrow> l) sequentially"
837     and "l' islimpt (range f)"
838   shows "l' = l"
839 proof (rule ccontr)
840   assume "l' \<noteq> l"
841   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
842     using hausdorff [OF \<open>l' \<noteq> l\<close>] by auto
843   have "eventually (\<lambda>n. f n \<in> t) sequentially"
844     using assms(1) \<open>open t\<close> \<open>l \<in> t\<close> by (rule topological_tendstoD)
845   then obtain N where "\<forall>n\<ge>N. f n \<in> t"
846     unfolding eventually_sequentially by auto
848   have "UNIV = {..<N} \<union> {N..}"
849     by auto
850   then have "l' islimpt (f ` ({..<N} \<union> {N..}))"
851     using assms(2) by simp
852   then have "l' islimpt (f ` {..<N} \<union> f ` {N..})"
854   then have "l' islimpt (f ` {N..})"
856   then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
857     using \<open>l' \<in> s\<close> \<open>open s\<close> by (rule islimptE)
858   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"
859     by auto
860   with \<open>\<forall>n\<ge>N. f n \<in> t\<close> have "f n \<in> s \<inter> t"
861     by simp
862   with \<open>s \<inter> t = {}\<close> show False
863     by simp
864 qed
867 subsection \<open>Interior of a Set\<close>
869 definition%important interior :: "('a::topological_space) set \<Rightarrow> 'a set" where
870 "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
872 lemma interiorI [intro?]:
873   assumes "open T" and "x \<in> T" and "T \<subseteq> S"
874   shows "x \<in> interior S"
875   using assms unfolding interior_def by fast
877 lemma interiorE [elim?]:
878   assumes "x \<in> interior S"
879   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
880   using assms unfolding interior_def by fast
882 lemma open_interior [simp, intro]: "open (interior S)"
883   by (simp add: interior_def open_Union)
885 lemma interior_subset: "interior S \<subseteq> S"
886   by (auto simp: interior_def)
888 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
889   by (auto simp: interior_def)
891 lemma interior_open: "open S \<Longrightarrow> interior S = S"
892   by (intro equalityI interior_subset interior_maximal subset_refl)
894 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
895   by (metis open_interior interior_open)
897 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
898   by (metis interior_maximal interior_subset subset_trans)
900 lemma interior_empty [simp]: "interior {} = {}"
901   using open_empty by (rule interior_open)
903 lemma interior_UNIV [simp]: "interior UNIV = UNIV"
904   using open_UNIV by (rule interior_open)
906 lemma interior_interior [simp]: "interior (interior S) = interior S"
907   using open_interior by (rule interior_open)
909 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
910   by (auto simp: interior_def)
912 lemma interior_unique:
913   assumes "T \<subseteq> S" and "open T"
914   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
915   shows "interior S = T"
916   by (intro equalityI assms interior_subset open_interior interior_maximal)
918 lemma interior_singleton [simp]: "interior {a} = {}"
919   for a :: "'a::perfect_space"
920   apply (rule interior_unique, simp_all)
921   using not_open_singleton subset_singletonD
922   apply fastforce
923   done
925 lemma interior_Int [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
926   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
927     Int_lower2 interior_maximal interior_subset open_Int open_interior)
929 lemma eventually_nhds_in_nhd: "x \<in> interior s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
930   using interior_subset[of s] by (subst eventually_nhds) blast
932 lemma interior_limit_point [intro]:
933   fixes x :: "'a::perfect_space"
934   assumes x: "x \<in> interior S"
935   shows "x islimpt S"
936   using x islimpt_UNIV [of x]
937   unfolding interior_def islimpt_def
938   apply (clarsimp, rename_tac T T')
939   apply (drule_tac x="T \<inter> T'" in spec)
940   apply (auto simp: open_Int)
941   done
943 lemma interior_closed_Un_empty_interior:
944   assumes cS: "closed S"
945     and iT: "interior T = {}"
946   shows "interior (S \<union> T) = interior S"
947 proof
948   show "interior S \<subseteq> interior (S \<union> T)"
949     by (rule interior_mono) (rule Un_upper1)
950   show "interior (S \<union> T) \<subseteq> interior S"
951   proof
952     fix x
953     assume "x \<in> interior (S \<union> T)"
954     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
955     show "x \<in> interior S"
956     proof (rule ccontr)
957       assume "x \<notin> interior S"
958       with \<open>x \<in> R\<close> \<open>open R\<close> obtain y where "y \<in> R - S"
959         unfolding interior_def by fast
960       from \<open>open R\<close> \<open>closed S\<close> have "open (R - S)"
961         by (rule open_Diff)
962       from \<open>R \<subseteq> S \<union> T\<close> have "R - S \<subseteq> T"
963         by fast
964       from \<open>y \<in> R - S\<close> \<open>open (R - S)\<close> \<open>R - S \<subseteq> T\<close> \<open>interior T = {}\<close> show False
965         unfolding interior_def by fast
966     qed
967   qed
968 qed
970 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
971 proof (rule interior_unique)
972   show "interior A \<times> interior B \<subseteq> A \<times> B"
973     by (intro Sigma_mono interior_subset)
974   show "open (interior A \<times> interior B)"
975     by (intro open_Times open_interior)
976   fix T
977   assume "T \<subseteq> A \<times> B" and "open T"
978   then show "T \<subseteq> interior A \<times> interior B"
979   proof safe
980     fix x y
981     assume "(x, y) \<in> T"
982     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
983       using \<open>open T\<close> unfolding open_prod_def by fast
984     then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
985       using \<open>T \<subseteq> A \<times> B\<close> by auto
986     then show "x \<in> interior A" and "y \<in> interior B"
987       by (auto intro: interiorI)
988   qed
989 qed
991 lemma interior_Ici:
992   fixes x :: "'a :: {dense_linorder,linorder_topology}"
993   assumes "b < x"
994   shows "interior {x ..} = {x <..}"
995 proof (rule interior_unique)
996   fix T
997   assume "T \<subseteq> {x ..}" "open T"
998   moreover have "x \<notin> T"
999   proof
1000     assume "x \<in> T"
1001     obtain y where "y < x" "{y <.. x} \<subseteq> T"
1002       using open_left[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>b < x\<close>] by auto
1003     with dense[OF \<open>y < x\<close>] obtain z where "z \<in> T" "z < x"
1004       by (auto simp: subset_eq Ball_def)
1005     with \<open>T \<subseteq> {x ..}\<close> show False by auto
1006   qed
1007   ultimately show "T \<subseteq> {x <..}"
1008     by (auto simp: subset_eq less_le)
1009 qed auto
1011 lemma interior_Iic:
1012   fixes x :: "'a ::{dense_linorder,linorder_topology}"
1013   assumes "x < b"
1014   shows "interior {.. x} = {..< x}"
1015 proof (rule interior_unique)
1016   fix T
1017   assume "T \<subseteq> {.. x}" "open T"
1018   moreover have "x \<notin> T"
1019   proof
1020     assume "x \<in> T"
1021     obtain y where "x < y" "{x ..< y} \<subseteq> T"
1022       using open_right[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>x < b\<close>] by auto
1023     with dense[OF \<open>x < y\<close>] obtain z where "z \<in> T" "x < z"
1024       by (auto simp: subset_eq Ball_def less_le)
1025     with \<open>T \<subseteq> {.. x}\<close> show False by auto
1026   qed
1027   ultimately show "T \<subseteq> {..< x}"
1028     by (auto simp: subset_eq less_le)
1029 qed auto
1031 lemma countable_disjoint_nonempty_interior_subsets:
1032   fixes \<F> :: "'a::second_countable_topology set set"
1033   assumes pw: "pairwise disjnt \<F>" and int: "\<And>S. \<lbrakk>S \<in> \<F>; interior S = {}\<rbrakk> \<Longrightarrow> S = {}"
1034   shows "countable \<F>"
1035 proof (rule countable_image_inj_on)
1036   have "disjoint (interior ` \<F>)"
1037     using pw by (simp add: disjoint_image_subset interior_subset)
1038   then show "countable (interior ` \<F>)"
1039     by (auto intro: countable_disjoint_open_subsets)
1040   show "inj_on interior \<F>"
1041     using pw apply (clarsimp simp: inj_on_def pairwise_def)
1042     apply (metis disjnt_def disjnt_subset1 inf.orderE int interior_subset)
1043     done
1044 qed
1046 subsection \<open>Closure of a Set\<close>
1048 definition%important closure :: "('a::topological_space) set \<Rightarrow> 'a set" where
1049 "closure S = S \<union> {x . x islimpt S}"
1051 lemma interior_closure: "interior S = - (closure (- S))"
1052   by (auto simp: interior_def closure_def islimpt_def)
1054 lemma closure_interior: "closure S = - interior (- S)"
1057 lemma closed_closure[simp, intro]: "closed (closure S)"
1058   by (simp add: closure_interior closed_Compl)
1060 lemma closure_subset: "S \<subseteq> closure S"
1063 lemma closure_hull: "closure S = closed hull S"
1064   by (auto simp: hull_def closure_interior interior_def)
1066 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
1067   unfolding closure_hull using closed_Inter by (rule hull_eq)
1069 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
1070   by (simp only: closure_eq)
1072 lemma closure_closure [simp]: "closure (closure S) = closure S"
1073   unfolding closure_hull by (rule hull_hull)
1075 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
1076   unfolding closure_hull by (rule hull_mono)
1078 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
1079   unfolding closure_hull by (rule hull_minimal)
1081 lemma closure_unique:
1082   assumes "S \<subseteq> T"
1083     and "closed T"
1084     and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
1085   shows "closure S = T"
1086   using assms unfolding closure_hull by (rule hull_unique)
1088 lemma closure_empty [simp]: "closure {} = {}"
1089   using closed_empty by (rule closure_closed)
1091 lemma closure_UNIV [simp]: "closure UNIV = UNIV"
1092   using closed_UNIV by (rule closure_closed)
1094 lemma closure_Un [simp]: "closure (S \<union> T) = closure S \<union> closure T"
1097 lemma closure_eq_empty [iff]: "closure S = {} \<longleftrightarrow> S = {}"
1098   using closure_empty closure_subset[of S] by blast
1100 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
1101   using closure_eq[of S] closure_subset[of S] by simp
1103 lemma open_Int_closure_eq_empty: "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
1104   using open_subset_interior[of S "- T"]
1105   using interior_subset[of "- T"]
1106   by (auto simp: closure_interior)
1108 lemma open_Int_closure_subset: "open S \<Longrightarrow> S \<inter> closure T \<subseteq> closure (S \<inter> T)"
1109 proof
1110   fix x
1111   assume *: "open S" "x \<in> S \<inter> closure T"
1112   have "x islimpt (S \<inter> T)" if **: "x islimpt T"
1113   proof (rule islimptI)
1114     fix A
1115     assume "x \<in> A" "open A"
1116     with * have "x \<in> A \<inter> S" "open (A \<inter> S)"
1118     with ** obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
1119       by (rule islimptE)
1120     then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
1121       by simp_all
1122     then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
1123   qed
1124   with * show "x \<in> closure (S \<inter> T)"
1125     unfolding closure_def by blast
1126 qed
1128 lemma closure_complement: "closure (- S) = - interior S"
1131 lemma interior_complement: "interior (- S) = - closure S"
1134 lemma interior_diff: "interior(S - T) = interior S - closure T"
1135   by (simp add: Diff_eq interior_complement)
1137 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
1138 proof (rule closure_unique)
1139   show "A \<times> B \<subseteq> closure A \<times> closure B"
1140     by (intro Sigma_mono closure_subset)
1141   show "closed (closure A \<times> closure B)"
1142     by (intro closed_Times closed_closure)
1143   fix T
1144   assume "A \<times> B \<subseteq> T" and "closed T"
1145   then show "closure A \<times> closure B \<subseteq> T"
1146     apply (simp add: closed_def open_prod_def, clarify)
1147     apply (rule ccontr)
1148     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
1149     apply (simp add: closure_interior interior_def)
1150     apply (drule_tac x=C in spec)
1151     apply (drule_tac x=D in spec, auto)
1152     done
1153 qed
1155 lemma islimpt_in_closure: "(x islimpt S) = (x\<in>closure(S-{x}))"
1156   unfolding closure_def using islimpt_punctured by blast
1158 lemma connected_imp_connected_closure: "connected S \<Longrightarrow> connected (closure S)"
1159   by (rule connectedI) (meson closure_subset open_Int open_Int_closure_eq_empty subset_trans connectedD)
1161 lemma bdd_below_closure:
1162   fixes A :: "real set"
1163   assumes "bdd_below A"
1164   shows "bdd_below (closure A)"
1165 proof -
1166   from assms obtain m where "\<And>x. x \<in> A \<Longrightarrow> m \<le> x"
1167     by (auto simp: bdd_below_def)
1168   then have "A \<subseteq> {m..}" by auto
1169   then have "closure A \<subseteq> {m..}"
1170     using closed_real_atLeast by (rule closure_minimal)
1171   then show ?thesis
1172     by (auto simp: bdd_below_def)
1173 qed
1176 subsection \<open>Frontier (also known as boundary)\<close>
1178 definition%important frontier :: "('a::topological_space) set \<Rightarrow> 'a set" where
1179 "frontier S = closure S - interior S"
1181 lemma frontier_closed [iff]: "closed (frontier S)"
1182   by (simp add: frontier_def closed_Diff)
1184 lemma frontier_closures: "frontier S = closure S \<inter> closure (- S)"
1185   by (auto simp: frontier_def interior_closure)
1187 lemma frontier_Int: "frontier(S \<inter> T) = closure(S \<inter> T) \<inter> (frontier S \<union> frontier T)"
1188 proof -
1189   have "closure (S \<inter> T) \<subseteq> closure S" "closure (S \<inter> T) \<subseteq> closure T"
1191   then show ?thesis
1192     by (auto simp: frontier_closures)
1193 qed
1195 lemma frontier_Int_subset: "frontier(S \<inter> T) \<subseteq> frontier S \<union> frontier T"
1196   by (auto simp: frontier_Int)
1198 lemma frontier_Int_closed:
1199   assumes "closed S" "closed T"
1200   shows "frontier(S \<inter> T) = (frontier S \<inter> T) \<union> (S \<inter> frontier T)"
1201 proof -
1202   have "closure (S \<inter> T) = T \<inter> S"
1203     using assms by (simp add: Int_commute closed_Int)
1204   moreover have "T \<inter> (closure S \<inter> closure (- S)) = frontier S \<inter> T"
1205     by (simp add: Int_commute frontier_closures)
1206   ultimately show ?thesis
1207     by (simp add: Int_Un_distrib Int_assoc Int_left_commute assms frontier_closures)
1208 qed
1210 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
1211   by (metis frontier_def closure_closed Diff_subset)
1213 lemma frontier_empty [simp]: "frontier {} = {}"
1216 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
1217 proof -
1218   {
1219     assume "frontier S \<subseteq> S"
1220     then have "closure S \<subseteq> S"
1221       using interior_subset unfolding frontier_def by auto
1222     then have "closed S"
1223       using closure_subset_eq by auto
1224   }
1225   then show ?thesis using frontier_subset_closed[of S] ..
1226 qed
1228 lemma frontier_complement [simp]: "frontier (- S) = frontier S"
1229   by (auto simp: frontier_def closure_complement interior_complement)
1231 lemma frontier_Un_subset: "frontier(S \<union> T) \<subseteq> frontier S \<union> frontier T"
1232   by (metis compl_sup frontier_Int_subset frontier_complement)
1234 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
1235   using frontier_complement frontier_subset_eq[of "- S"]
1236   unfolding open_closed by auto
1238 lemma frontier_UNIV [simp]: "frontier UNIV = {}"
1239   using frontier_complement frontier_empty by fastforce
1241 lemma frontier_interiors: "frontier s = - interior(s) - interior(-s)"
1242   by (simp add: Int_commute frontier_def interior_closure)
1244 lemma frontier_interior_subset: "frontier(interior S) \<subseteq> frontier S"
1245   by (simp add: Diff_mono frontier_interiors interior_mono interior_subset)
1247 lemma closure_Un_frontier: "closure S = S \<union> frontier S"
1248 proof -
1249   have "S \<union> interior S = S"
1250     using interior_subset by auto
1251   then show ?thesis
1252     using closure_subset by (auto simp: frontier_def)
1253 qed
1256 subsection%unimportant \<open>Filters and the ``eventually true'' quantifier\<close>
1258 text \<open>Identify Trivial limits, where we can't approach arbitrarily closely.\<close>
1260 lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
1261 proof
1262   assume "trivial_limit (at a within S)"
1263   then show "\<not> a islimpt S"
1264     unfolding trivial_limit_def
1265     unfolding eventually_at_topological
1266     unfolding islimpt_def
1267     apply (clarsimp simp add: set_eq_iff)
1268     apply (rename_tac T, rule_tac x=T in exI)
1269     apply (clarsimp, drule_tac x=y in bspec, simp_all)
1270     done
1271 next
1272   assume "\<not> a islimpt S"
1273   then show "trivial_limit (at a within S)"
1274     unfolding trivial_limit_def eventually_at_topological islimpt_def
1275     by metis
1276 qed
1278 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
1279   using trivial_limit_within [of a UNIV] by simp
1281 lemma trivial_limit_at: "\<not> trivial_limit (at a)"
1282   for a :: "'a::perfect_space"
1283   by (rule at_neq_bot)
1285 lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"
1286   using islimpt_in_closure by (metis trivial_limit_within)
1288 lemma not_in_closure_trivial_limitI:
1289   "x \<notin> closure s \<Longrightarrow> trivial_limit (at x within s)"
1290   using not_trivial_limit_within[of x s]
1291   by safe (metis Diff_empty Diff_insert0 closure_subset contra_subsetD)
1293 lemma filterlim_at_within_closure_implies_filterlim: "filterlim f l (at x within s)"
1294   if "x \<in> closure s \<Longrightarrow> filterlim f l (at x within s)"
1295   by (metis bot.extremum filterlim_filtercomap filterlim_mono not_in_closure_trivial_limitI that)
1297 lemma at_within_eq_bot_iff: "at c within A = bot \<longleftrightarrow> c \<notin> closure (A - {c})"
1298   using not_trivial_limit_within[of c A] by blast
1300 text \<open>Some property holds "sufficiently close" to the limit point.\<close>
1302 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
1303   by simp
1305 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
1308 lemma Lim_topological:
1309   "(f \<longlongrightarrow> l) net \<longleftrightarrow>
1310     trivial_limit net \<or> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
1311   unfolding tendsto_def trivial_limit_eq by auto
1313 lemma eventually_within_Un:
1314   "eventually P (at x within (s \<union> t)) \<longleftrightarrow>
1315     eventually P (at x within s) \<and> eventually P (at x within t)"
1316   unfolding eventually_at_filter
1317   by (auto elim!: eventually_rev_mp)
1319 lemma Lim_within_union:
1320  "(f \<longlongrightarrow> l) (at x within (s \<union> t)) \<longleftrightarrow>
1321   (f \<longlongrightarrow> l) (at x within s) \<and> (f \<longlongrightarrow> l) (at x within t)"
1322   unfolding tendsto_def
1323   by (auto simp: eventually_within_Un)
1326 subsection \<open>Limits\<close>
1328 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f \<longlongrightarrow> l) net"
1329   by (rule topological_tendstoI) (auto elim: eventually_mono)
1331 text \<open>The expected monotonicity property.\<close>
1333 lemma Lim_Un:
1334   assumes "(f \<longlongrightarrow> l) (at x within S)" "(f \<longlongrightarrow> l) (at x within T)"
1335   shows "(f \<longlongrightarrow> l) (at x within (S \<union> T))"
1336   using assms unfolding at_within_union by (rule filterlim_sup)
1338 lemma Lim_Un_univ:
1339   "(f \<longlongrightarrow> l) (at x within S) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within T) \<Longrightarrow>
1340     S \<union> T = UNIV \<Longrightarrow> (f \<longlongrightarrow> l) (at x)"
1341   by (metis Lim_Un)
1343 text \<open>Interrelations between restricted and unrestricted limits.\<close>
1345 lemma Lim_at_imp_Lim_at_within: "(f \<longlongrightarrow> l) (at x) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S)"
1346   by (metis order_refl filterlim_mono subset_UNIV at_le)
1348 lemma eventually_within_interior:
1349   assumes "x \<in> interior S"
1350   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)"
1351   (is "?lhs = ?rhs")
1352 proof
1353   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
1354   {
1355     assume ?lhs
1356     then obtain A where "open A" and "x \<in> A" and "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
1357       by (auto simp: eventually_at_topological)
1358     with T have "open (A \<inter> T)" and "x \<in> A \<inter> T" and "\<forall>y \<in> A \<inter> T. y \<noteq> x \<longrightarrow> P y"
1359       by auto
1360     then show ?rhs
1361       by (auto simp: eventually_at_topological)
1362   next
1363     assume ?rhs
1364     then show ?lhs
1365       by (auto elim: eventually_mono simp: eventually_at_filter)
1366   }
1367 qed
1369 lemma at_within_interior: "x \<in> interior S \<Longrightarrow> at x within S = at x"
1370   unfolding filter_eq_iff by (intro allI eventually_within_interior)
1372 lemma Lim_within_LIMSEQ:
1373   fixes a :: "'a::first_countable_topology"
1374   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
1375   shows "(X \<longlongrightarrow> L) (at a within T)"
1376   using assms unfolding tendsto_def [where l=L]
1379 lemma Lim_right_bound:
1380   fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>
1381     'b::{linorder_topology, conditionally_complete_linorder}"
1382   assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
1383     and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
1384   shows "(f \<longlongrightarrow> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
1385 proof (cases "{x<..} \<inter> I = {}")
1386   case True
1387   then show ?thesis by simp
1388 next
1389   case False
1390   show ?thesis
1391   proof (rule order_tendstoI)
1392     fix a
1393     assume a: "a < Inf (f ` ({x<..} \<inter> I))"
1394     {
1395       fix y
1396       assume "y \<in> {x<..} \<inter> I"
1397       with False bnd have "Inf (f ` ({x<..} \<inter> I)) \<le> f y"
1398         by (auto intro!: cInf_lower bdd_belowI2)
1399       with a have "a < f y"
1400         by (blast intro: less_le_trans)
1401     }
1402     then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"
1403       by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)
1404   next
1405     fix a
1406     assume "Inf (f ` ({x<..} \<inter> I)) < a"
1407     from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"
1408       by auto
1409     then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"
1410       unfolding eventually_at_right[OF \<open>x < y\<close>] by (metis less_imp_le le_less_trans mono)
1411     then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"
1412       unfolding eventually_at_filter by eventually_elim simp
1413   qed
1414 qed
1416 (*could prove directly from islimpt_sequential_inj, but only for metric spaces*)
1417 lemma islimpt_sequential:
1418   fixes x :: "'a::first_countable_topology"
1419   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f \<longlongrightarrow> x) sequentially)"
1420     (is "?lhs = ?rhs")
1421 proof
1422   assume ?lhs
1423   from countable_basis_at_decseq[of x] obtain A where A:
1424       "\<And>i. open (A i)"
1425       "\<And>i. x \<in> A i"
1426       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
1427     by blast
1428   define f where "f n = (SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y)" for n
1429   {
1430     fix n
1431     from \<open>?lhs\<close> have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
1432       unfolding islimpt_def using A(1,2)[of n] by auto
1433     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"
1434       unfolding f_def by (rule someI_ex)
1435     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto
1436   }
1437   then have "\<forall>n. f n \<in> S - {x}" by auto
1438   moreover have "(\<lambda>n. f n) \<longlonglongrightarrow> x"
1439   proof (rule topological_tendstoI)
1440     fix S
1441     assume "open S" "x \<in> S"
1442     from A(3)[OF this] \<open>\<And>n. f n \<in> A n\<close>
1443     show "eventually (\<lambda>x. f x \<in> S) sequentially"
1444       by (auto elim!: eventually_mono)
1445   qed
1446   ultimately show ?rhs by fast
1447 next
1448   assume ?rhs
1449   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f \<longlonglongrightarrow> x"
1450     by auto
1451   show ?lhs
1452     unfolding islimpt_def
1453   proof safe
1454     fix T
1455     assume "open T" "x \<in> T"
1456     from lim[THEN topological_tendstoD, OF this] f
1457     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
1458       unfolding eventually_sequentially by auto
1459   qed
1460 qed
1462 text\<open>Deducing things about the limit from the elements.\<close>
1464 lemma Lim_in_closed_set:
1465   assumes "closed S"
1466     and "eventually (\<lambda>x. f(x) \<in> S) net"
1467     and "\<not> trivial_limit net" "(f \<longlongrightarrow> l) net"
1468   shows "l \<in> S"
1469 proof (rule ccontr)
1470   assume "l \<notin> S"
1471   with \<open>closed S\<close> have "open (- S)" "l \<in> - S"
1473   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
1474     by (rule topological_tendstoD)
1475   with assms(2) have "eventually (\<lambda>x. False) net"
1476     by (rule eventually_elim2) simp
1477   with assms(3) show "False"
1479 qed
1481 text\<open>These are special for limits out of the same topological space.\<close>
1483 lemma Lim_within_id: "(id \<longlongrightarrow> a) (at a within s)"
1484   unfolding id_def by (rule tendsto_ident_at)
1486 lemma Lim_at_id: "(id \<longlongrightarrow> a) (at a)"
1487   unfolding id_def by (rule tendsto_ident_at)
1489 text\<open>It's also sometimes useful to extract the limit point from the filter.\<close>
1491 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"
1492   where "netlimit F \<equiv> Lim F (\<lambda>x. x)"
1494 lemma netlimit_within: "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"
1495   by (rule tendsto_Lim) (auto intro: tendsto_intros)
1497 lemma netlimit_at [simp]:
1498   fixes a :: "'a::{perfect_space,t2_space}"
1499   shows "netlimit (at a) = a"
1500   using netlimit_within [of a UNIV] by simp
1502 lemma lim_within_interior:
1503   "x \<in> interior S \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S) \<longleftrightarrow> (f \<longlongrightarrow> l) (at x)"
1504   by (metis at_within_interior)
1506 lemma netlimit_within_interior:
1507   fixes x :: "'a::{t2_space,perfect_space}"
1508   assumes "x \<in> interior S"
1509   shows "netlimit (at x within S) = x"
1510   using assms by (metis at_within_interior netlimit_at)
1512 text\<open>Useful lemmas on closure and set of possible sequential limits.\<close>
1514 lemma closure_sequential:
1515   fixes l :: "'a::first_countable_topology"
1516   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially)"
1517   (is "?lhs = ?rhs")
1518 proof
1519   assume "?lhs"
1520   moreover
1521   {
1522     assume "l \<in> S"
1523     then have "?rhs" using tendsto_const[of l sequentially] by auto
1524   }
1525   moreover
1526   {
1527     assume "l islimpt S"
1528     then have "?rhs" unfolding islimpt_sequential by auto
1529   }
1530   ultimately show "?rhs"
1531     unfolding closure_def by auto
1532 next
1533   assume "?rhs"
1534   then show "?lhs" unfolding closure_def islimpt_sequential by auto
1535 qed
1537 lemma closed_sequential_limits:
1538   fixes S :: "'a::first_countable_topology set"
1539   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially \<longrightarrow> l \<in> S)"
1540 by (metis closure_sequential closure_subset_eq subset_iff)
1542 lemma tendsto_If_within_closures:
1543   assumes f: "x \<in> s \<union> (closure s \<inter> closure t) \<Longrightarrow>
1544       (f \<longlongrightarrow> l x) (at x within s \<union> (closure s \<inter> closure t))"
1545   assumes g: "x \<in> t \<union> (closure s \<inter> closure t) \<Longrightarrow>
1546       (g \<longlongrightarrow> l x) (at x within t \<union> (closure s \<inter> closure t))"
1547   assumes "x \<in> s \<union> t"
1548   shows "((\<lambda>x. if x \<in> s then f x else g x) \<longlongrightarrow> l x) (at x within s \<union> t)"
1549 proof -
1550   have *: "(s \<union> t) \<inter> {x. x \<in> s} = s" "(s \<union> t) \<inter> {x. x \<notin> s} = t - s"
1551     by auto
1552   have "(f \<longlongrightarrow> l x) (at x within s)"
1553     by (rule filterlim_at_within_closure_implies_filterlim)
1554        (use \<open>x \<in> _\<close> in \<open>auto simp: inf_commute closure_def intro: tendsto_within_subset[OF f]\<close>)
1555   moreover
1556   have "(g \<longlongrightarrow> l x) (at x within t - s)"
1557     by (rule filterlim_at_within_closure_implies_filterlim)
1558       (use \<open>x \<in> _\<close> in
1559         \<open>auto intro!: tendsto_within_subset[OF g] simp: closure_def intro: islimpt_subset\<close>)
1560   ultimately show ?thesis
1561     by (intro filterlim_at_within_If) (simp_all only: *)
1562 qed
1565 subsection \<open>Compactness\<close>
1567 lemma brouwer_compactness_lemma:
1568   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1569   assumes "compact s"
1570     and "continuous_on s f"
1571     and "\<not> (\<exists>x\<in>s. f x = 0)"
1572   obtains d where "0 < d" and "\<forall>x\<in>s. d \<le> norm (f x)"
1573 proof (cases "s = {}")
1574   case True
1575   show thesis
1576     by (rule that [of 1]) (auto simp: True)
1577 next
1578   case False
1579   have "continuous_on s (norm \<circ> f)"
1580     by (rule continuous_intros continuous_on_norm assms(2))+
1581   with False obtain x where x: "x \<in> s" "\<forall>y\<in>s. (norm \<circ> f) x \<le> (norm \<circ> f) y"
1582     using continuous_attains_inf[OF assms(1), of "norm \<circ> f"]
1583     unfolding o_def
1584     by auto
1585   have "(norm \<circ> f) x > 0"
1586     using assms(3) and x(1)
1587     by auto
1588   then show ?thesis
1589     by (rule that) (insert x(2), auto simp: o_def)
1590 qed
1592 subsubsection \<open>Bolzano-Weierstrass property\<close>
1594 proposition Heine_Borel_imp_Bolzano_Weierstrass:
1595   assumes "compact s"
1596     and "infinite t"
1597     and "t \<subseteq> s"
1598   shows "\<exists>x \<in> s. x islimpt t"
1599 proof (rule ccontr)
1600   assume "\<not> (\<exists>x \<in> s. x islimpt t)"
1601   then obtain f where f: "\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)"
1602     unfolding islimpt_def
1603     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]
1604     by auto
1605   obtain g where g: "g \<subseteq> {t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
1606     using assms(1)[unfolded compact_eq_Heine_Borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]
1607     using f by auto
1608   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa"
1609     by auto
1610   {
1611     fix x y
1612     assume "x \<in> t" "y \<in> t" "f x = f y"
1613     then have "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x"
1614       using f[THEN bspec[where x=x]] and \<open>t \<subseteq> s\<close> by auto
1615     then have "x = y"
1616       using \<open>f x = f y\<close> and f[THEN bspec[where x=y]] and \<open>y \<in> t\<close> and \<open>t \<subseteq> s\<close>
1617       by auto
1618   }
1619   then have "inj_on f t"
1620     unfolding inj_on_def by simp
1621   then have "infinite (f ` t)"
1622     using assms(2) using finite_imageD by auto
1623   moreover
1624   {
1625     fix x
1626     assume "x \<in> t" "f x \<notin> g"
1627     from g(3) assms(3) \<open>x \<in> t\<close> obtain h where "h \<in> g" and "x \<in> h"
1628       by auto
1629     then obtain y where "y \<in> s" "h = f y"
1630       using g'[THEN bspec[where x=h]] by auto
1631     then have "y = x"
1632       using f[THEN bspec[where x=y]] and \<open>x\<in>t\<close> and \<open>x\<in>h\<close>[unfolded \<open>h = f y\<close>]
1633       by auto
1634     then have False
1635       using \<open>f x \<notin> g\<close> \<open>h \<in> g\<close> unfolding \<open>h = f y\<close>
1636       by auto
1637   }
1638   then have "f ` t \<subseteq> g" by auto
1639   ultimately show False
1640     using g(2) using finite_subset by auto
1641 qed
1643 lemma sequence_infinite_lemma:
1644   fixes f :: "nat \<Rightarrow> 'a::t1_space"
1645   assumes "\<forall>n. f n \<noteq> l"
1646     and "(f \<longlongrightarrow> l) sequentially"
1647   shows "infinite (range f)"
1648 proof
1649   assume "finite (range f)"
1650   then have "closed (range f)"
1651     by (rule finite_imp_closed)
1652   then have "open (- range f)"
1653     by (rule open_Compl)
1654   from assms(1) have "l \<in> - range f"
1655     by auto
1656   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
1657     using \<open>open (- range f)\<close> \<open>l \<in> - range f\<close>
1658     by (rule topological_tendstoD)
1659   then show False
1660     unfolding eventually_sequentially
1661     by auto
1662 qed
1664 lemma Bolzano_Weierstrass_imp_closed:
1665   fixes s :: "'a::{first_countable_topology,t2_space} set"
1666   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
1667   shows "closed s"
1668 proof -
1669   {
1670     fix x l
1671     assume as: "\<forall>n::nat. x n \<in> s" "(x \<longlongrightarrow> l) sequentially"
1672     then have "l \<in> s"
1673     proof (cases "\<forall>n. x n \<noteq> l")
1674       case False
1675       then show "l\<in>s" using as(1) by auto
1676     next
1677       case True note cas = this
1678       with as(2) have "infinite (range x)"
1679         using sequence_infinite_lemma[of x l] by auto
1680       then obtain l' where "l'\<in>s" "l' islimpt (range x)"
1681         using assms[THEN spec[where x="range x"]] as(1) by auto
1682       then show "l\<in>s" using sequence_unique_limpt[of x l l']
1683         using as cas by auto
1684     qed
1685   }
1686   then show ?thesis
1687     unfolding closed_sequential_limits by fast
1688 qed
1690 lemma closure_insert:
1691   fixes x :: "'a::t1_space"
1692   shows "closure (insert x s) = insert x (closure s)"
1693   apply (rule closure_unique)
1694   apply (rule insert_mono [OF closure_subset])
1695   apply (rule closed_insert [OF closed_closure])
1697   done
1700 text\<open>In particular, some common special cases.\<close>
1702 lemma compact_Un [intro]:
1703   assumes "compact s"
1704     and "compact t"
1705   shows " compact (s \<union> t)"
1706 proof (rule compactI)
1707   fix f
1708   assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"
1709   from * \<open>compact s\<close> obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"
1710     unfolding compact_eq_Heine_Borel by (auto elim!: allE[of _ f])
1711   moreover
1712   from * \<open>compact t\<close> obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"
1713     unfolding compact_eq_Heine_Borel by (auto elim!: allE[of _ f])
1714   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"
1715     by (auto intro!: exI[of _ "s' \<union> t'"])
1716 qed
1718 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"
1719   by (induct set: finite) auto
1721 lemma compact_UN [intro]:
1722   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"
1723   by (rule compact_Union) auto
1725 lemma closed_Int_compact [intro]:
1726   assumes "closed s"
1727     and "compact t"
1728   shows "compact (s \<inter> t)"
1729   using compact_Int_closed [of t s] assms
1732 lemma compact_Int [intro]:
1733   fixes s t :: "'a :: t2_space set"
1734   assumes "compact s"
1735     and "compact t"
1736   shows "compact (s \<inter> t)"
1737   using assms by (intro compact_Int_closed compact_imp_closed)
1739 lemma compact_sing [simp]: "compact {a}"
1740   unfolding compact_eq_Heine_Borel by auto
1742 lemma compact_insert [simp]:
1743   assumes "compact s"
1744   shows "compact (insert x s)"
1745 proof -
1746   have "compact ({x} \<union> s)"
1747     using compact_sing assms by (rule compact_Un)
1748   then show ?thesis by simp
1749 qed
1751 lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"
1752   by (induct set: finite) simp_all
1754 lemma open_delete:
1755   fixes s :: "'a::t1_space set"
1756   shows "open s \<Longrightarrow> open (s - {x})"
1760 text\<open>Compactness expressed with filters\<close>
1762 lemma closure_iff_nhds_not_empty:
1763   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
1764 proof safe
1765   assume x: "x \<in> closure X"
1766   fix S A
1767   assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"
1768   then have "x \<notin> closure (-S)"
1769     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)
1770   with x have "x \<in> closure X - closure (-S)"
1771     by auto
1772   also have "\<dots> \<subseteq> closure (X \<inter> S)"
1773     using \<open>open S\<close> open_Int_closure_subset[of S X] by (simp add: closed_Compl ac_simps)
1774   finally have "X \<inter> S \<noteq> {}" by auto
1775   then show False using \<open>X \<inter> A = {}\<close> \<open>S \<subseteq> A\<close> by auto
1776 next
1777   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"
1778   from this[THEN spec, of "- X", THEN spec, of "- closure X"]
1779   show "x \<in> closure X"
1780     by (simp add: closure_subset open_Compl)
1781 qed
1783 lemma compact_filter:
1784   "compact U \<longleftrightarrow> (\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot))"
1785 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
1786   fix F
1787   assume "compact U"
1788   assume F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"
1789   then have "U \<noteq> {}"
1790     by (auto simp: eventually_False)
1792   define Z where "Z = closure ` {A. eventually (\<lambda>x. x \<in> A) F}"
1793   then have "\<forall>z\<in>Z. closed z"
1794     by auto
1795   moreover
1796   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"
1797     unfolding Z_def by (auto elim: eventually_mono intro: subsetD[OF closure_subset])
1798   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"
1799   proof (intro allI impI)
1800     fix B assume "finite B" "B \<subseteq> Z"
1801     with \<open>finite B\<close> ev_Z F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"
1802       by (auto simp: eventually_ball_finite_distrib eventually_conj_iff)
1803     with F show "U \<inter> \<Inter>B \<noteq> {}"
1804       by (intro notI) (simp add: eventually_False)
1805   qed
1806   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"
1807     using \<open>compact U\<close> unfolding compact_fip by blast
1808   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z"
1809     by auto
1811   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"
1812     unfolding eventually_inf eventually_nhds
1813   proof safe
1814     fix P Q R S
1815     assume "eventually R F" "open S" "x \<in> S"
1816     with open_Int_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]
1817     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)
1818     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"
1819     ultimately show False by (auto simp: set_eq_iff)
1820   qed
1821   with \<open>x \<in> U\<close> show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"
1822     by (metis eventually_bot)
1823 next
1824   fix A
1825   assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"
1826   define F where "F = (INF a\<in>insert U A. principal a)"
1827   have "F \<noteq> bot"
1828     unfolding F_def
1829   proof (rule INF_filter_not_bot)
1830     fix X
1831     assume X: "X \<subseteq> insert U A" "finite X"
1832     with A(2)[THEN spec, of "X - {U}"] have "U \<inter> \<Inter>(X - {U}) \<noteq> {}"
1833       by auto
1834     with X show "(INF a\<in>X. principal a) \<noteq> bot"
1835       by (auto simp: INF_principal_finite principal_eq_bot_iff)
1836   qed
1837   moreover
1838   have "F \<le> principal U"
1839     unfolding F_def by auto
1840   then have "eventually (\<lambda>x. x \<in> U) F"
1841     by (auto simp: le_filter_def eventually_principal)
1842   moreover
1843   assume "\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot)"
1844   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"
1845     by auto
1847   { fix V assume "V \<in> A"
1848     then have "F \<le> principal V"
1849       unfolding F_def by (intro INF_lower2[of V]) auto
1850     then have V: "eventually (\<lambda>x. x \<in> V) F"
1851       by (auto simp: le_filter_def eventually_principal)
1852     have "x \<in> closure V"
1853       unfolding closure_iff_nhds_not_empty
1854     proof (intro impI allI)
1855       fix S A
1856       assume "open S" "x \<in> S" "S \<subseteq> A"
1857       then have "eventually (\<lambda>x. x \<in> A) (nhds x)"
1858         by (auto simp: eventually_nhds)
1859       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"
1860         by (auto simp: eventually_inf)
1861       with x show "V \<inter> A \<noteq> {}"
1862         by (auto simp del: Int_iff simp add: trivial_limit_def)
1863     qed
1864     then have "x \<in> V"
1865       using \<open>V \<in> A\<close> A(1) by simp
1866   }
1867   with \<open>x\<in>U\<close> have "x \<in> U \<inter> \<Inter>A" by auto
1868   with \<open>U \<inter> \<Inter>A = {}\<close> show False by auto
1869 qed
1871 definition%important countably_compact :: "('a::topological_space) set \<Rightarrow> bool" where
1872 "countably_compact U \<longleftrightarrow>
1873   (\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A
1874      \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T))"
1876 lemma countably_compactE:
1877   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"
1878   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
1879   using assms unfolding countably_compact_def by metis
1881 lemma countably_compactI:
1882   assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union>C \<Longrightarrow> countable C \<Longrightarrow> (\<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union>C')"
1883   shows "countably_compact s"
1884   using assms unfolding countably_compact_def by metis
1886 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"
1887   by (auto simp: compact_eq_Heine_Borel countably_compact_def)
1889 lemma countably_compact_imp_compact:
1890   assumes "countably_compact U"
1891     and ccover: "countable B" "\<forall>b\<in>B. open b"
1892     and basis: "\<And>T x. open T \<Longrightarrow> x \<in> T \<Longrightarrow> x \<in> U \<Longrightarrow> \<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T"
1893   shows "compact U"
1894   using \<open>countably_compact U\<close>
1895   unfolding compact_eq_Heine_Borel countably_compact_def
1896 proof safe
1897   fix A
1898   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
1899   assume *: "\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
1900   moreover define C where "C = {b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"
1901   ultimately have "countable C" "\<forall>a\<in>C. open a"
1902     unfolding C_def using ccover by auto
1903   moreover
1904   have "\<Union>A \<inter> U \<subseteq> \<Union>C"
1905   proof safe
1906     fix x a
1907     assume "x \<in> U" "x \<in> a" "a \<in> A"
1908     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a"
1909       by blast
1910     with \<open>a \<in> A\<close> show "x \<in> \<Union>C"
1911       unfolding C_def by auto
1912   qed
1913   then have "U \<subseteq> \<Union>C" using \<open>U \<subseteq> \<Union>A\<close> by auto
1914   ultimately obtain T where T: "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"
1915     using * by metis
1916   then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"
1917     by (auto simp: C_def)
1918   then obtain f where "\<forall>t\<in>T. f t \<in> A \<and> t \<inter> U \<subseteq> f t"
1919     unfolding bchoice_iff Bex_def ..
1920   with T show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
1921     unfolding C_def by (intro exI[of _ "f`T"]) fastforce
1922 qed
1924 proposition countably_compact_imp_compact_second_countable:
1925   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"
1926 proof (rule countably_compact_imp_compact)
1927   fix T and x :: 'a
1928   assume "open T" "x \<in> T"
1929   from topological_basisE[OF is_basis this] obtain b where
1930     "b \<in> (SOME B. countable B \<and> topological_basis B)" "x \<in> b" "b \<subseteq> T" .
1931   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T"
1932     by blast
1933 qed (insert countable_basis topological_basis_open[OF is_basis], auto)
1935 lemma countably_compact_eq_compact:
1936   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"
1937   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast
1939 subsubsection\<open>Sequential compactness\<close>
1941 definition%important seq_compact :: "'a::topological_space set \<Rightarrow> bool" where
1942 "seq_compact S \<longleftrightarrow>
1943   (\<forall>f. (\<forall>n. f n \<in> S)
1944     \<longrightarrow> (\<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially))"
1946 lemma seq_compactI:
1947   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
1948   shows "seq_compact S"
1949   unfolding seq_compact_def using assms by fast
1951 lemma seq_compactE:
1952   assumes "seq_compact S" "\<forall>n. f n \<in> S"
1953   obtains l r where "l \<in> S" "strict_mono (r :: nat \<Rightarrow> nat)" "((f \<circ> r) \<longlongrightarrow> l) sequentially"
1954   using assms unfolding seq_compact_def by fast
1956 lemma closed_sequentially: (* TODO: move upwards *)
1957   assumes "closed s" and "\<forall>n. f n \<in> s" and "f \<longlonglongrightarrow> l"
1958   shows "l \<in> s"
1959 proof (rule ccontr)
1960   assume "l \<notin> s"
1961   with \<open>closed s\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "eventually (\<lambda>n. f n \<in> - s) sequentially"
1962     by (fast intro: topological_tendstoD)
1963   with \<open>\<forall>n. f n \<in> s\<close> show "False"
1964     by simp
1965 qed
1967 lemma seq_compact_Int_closed:
1968   assumes "seq_compact s" and "closed t"
1969   shows "seq_compact (s \<inter> t)"
1970 proof (rule seq_compactI)
1971   fix f assume "\<forall>n::nat. f n \<in> s \<inter> t"
1972   hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"
1973     by simp_all
1974   from \<open>seq_compact s\<close> and \<open>\<forall>n. f n \<in> s\<close>
1975   obtain l r where "l \<in> s" and r: "strict_mono r" and l: "(f \<circ> r) \<longlonglongrightarrow> l"
1976     by (rule seq_compactE)
1977   from \<open>\<forall>n. f n \<in> t\<close> have "\<forall>n. (f \<circ> r) n \<in> t"
1978     by simp
1979   from \<open>closed t\<close> and this and l have "l \<in> t"
1980     by (rule closed_sequentially)
1981   with \<open>l \<in> s\<close> and r and l show "\<exists>l\<in>s \<inter> t. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
1982     by fast
1983 qed
1985 lemma seq_compact_closed_subset:
1986   assumes "closed s" and "s \<subseteq> t" and "seq_compact t"
1987   shows "seq_compact s"
1988   using assms seq_compact_Int_closed [of t s] by (simp add: Int_absorb1)
1990 lemma seq_compact_imp_countably_compact:
1991   fixes U :: "'a :: first_countable_topology set"
1992   assumes "seq_compact U"
1993   shows "countably_compact U"
1994 proof (safe intro!: countably_compactI)
1995   fix A
1996   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"
1997   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> strict_mono (r :: nat \<Rightarrow> nat) \<and> (X \<circ> r) \<longlonglongrightarrow> x"
1998     using \<open>seq_compact U\<close> by (fastforce simp: seq_compact_def subset_eq)
1999   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
2000   proof cases
2001     assume "finite A"
2002     with A show ?thesis by auto
2003   next
2004     assume "infinite A"
2005     then have "A \<noteq> {}" by auto
2006     show ?thesis
2007     proof (rule ccontr)
2008       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
2009       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)"
2010         by auto
2011       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T"
2012         by metis
2013       define X where "X n = X' (from_nat_into A ` {.. n})" for n
2014       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"
2015         using \<open>A \<noteq> {}\<close> unfolding X_def by (intro T) (auto intro: from_nat_into)
2016       then have "range X \<subseteq> U"
2017         by auto
2018       with subseq[of X] obtain r x where "x \<in> U" and r: "strict_mono r" "(X \<circ> r) \<longlonglongrightarrow> x"
2019         by auto
2020       from \<open>x\<in>U\<close> \<open>U \<subseteq> \<Union>A\<close> from_nat_into_surj[OF \<open>countable A\<close>]
2021       obtain n where "x \<in> from_nat_into A n" by auto
2022       with r(2) A(1) from_nat_into[OF \<open>A \<noteq> {}\<close>, of n]
2023       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"
2024         unfolding tendsto_def by (auto simp: comp_def)
2025       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"
2026         by (auto simp: eventually_sequentially)
2027       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"
2028         by auto
2029       moreover from \<open>strict_mono r\<close>[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"
2030         by (auto intro!: exI[of _ "max n N"])
2031       ultimately show False
2032         by auto
2033     qed
2034   qed
2035 qed
2037 lemma compact_imp_seq_compact:
2038   fixes U :: "'a :: first_countable_topology set"
2039   assumes "compact U"
2040   shows "seq_compact U"
2041   unfolding seq_compact_def
2042 proof safe
2043   fix X :: "nat \<Rightarrow> 'a"
2044   assume "\<forall>n. X n \<in> U"
2045   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"
2046     by (auto simp: eventually_filtermap)
2047   moreover
2048   have "filtermap X sequentially \<noteq> bot"
2049     by (simp add: trivial_limit_def eventually_filtermap)
2050   ultimately
2051   obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")
2052     using \<open>compact U\<close> by (auto simp: compact_filter)
2054   from countable_basis_at_decseq[of x]
2055   obtain A where A:
2056       "\<And>i. open (A i)"
2057       "\<And>i. x \<in> A i"
2058       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
2059     by blast
2060   define s where "s n i = (SOME j. i < j \<and> X j \<in> A (Suc n))" for n i
2061   {
2062     fix n i
2063     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"
2064     proof (rule ccontr)
2065       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"
2066       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)"
2067         by auto
2068       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"
2069         by (auto simp: eventually_filtermap eventually_sequentially)
2070       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"
2071         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)
2072       ultimately have "eventually (\<lambda>x. False) ?F"
2073         by (auto simp: eventually_inf)
2074       with x show False
2076     qed
2077     then have "i < s n i" "X (s n i) \<in> A (Suc n)"
2078       unfolding s_def by (auto intro: someI2_ex)
2079   }
2080   note s = this
2081   define r where "r = rec_nat (s 0 0) s"
2082   have "strict_mono r"
2083     by (auto simp: r_def s strict_mono_Suc_iff)
2084   moreover
2085   have "(\<lambda>n. X (r n)) \<longlonglongrightarrow> x"
2086   proof (rule topological_tendstoI)
2087     fix S
2088     assume "open S" "x \<in> S"
2089     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
2090       by auto
2091     moreover
2092     {
2093       fix i
2094       assume "Suc 0 \<le> i"
2095       then have "X (r i) \<in> A i"
2096         by (cases i) (simp_all add: r_def s)
2097     }
2098     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially"
2099       by (auto simp: eventually_sequentially)
2100     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"
2101       by eventually_elim auto
2102   qed
2103   ultimately show "\<exists>x \<in> U. \<exists>r. strict_mono r \<and> (X \<circ> r) \<longlonglongrightarrow> x"
2104     using \<open>x \<in> U\<close> by (auto simp: convergent_def comp_def)
2105 qed
2107 lemma countably_compact_imp_acc_point:
2108   assumes "countably_compact s"
2109     and "countable t"
2110     and "infinite t"
2111     and "t \<subseteq> s"
2112   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"
2113 proof (rule ccontr)
2114   define C where "C = (\<lambda>F. interior (F \<union> (- t))) ` {F. finite F \<and> F \<subseteq> t }"
2115   note \<open>countably_compact s\<close>
2116   moreover have "\<forall>t\<in>C. open t"
2117     by (auto simp: C_def)
2118   moreover
2119   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
2120   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis
2121   have "s \<subseteq> \<Union>C"
2122     using \<open>t \<subseteq> s\<close>
2123     unfolding C_def
2124     apply (safe dest!: s)
2125     apply (rule_tac a="U \<inter> t" in UN_I)
2126     apply (auto intro!: interiorI simp add: finite_subset)
2127     done
2128   moreover
2129   from \<open>countable t\<close> have "countable C"
2130     unfolding C_def by (auto intro: countable_Collect_finite_subset)
2131   ultimately
2132   obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> \<Union>D"
2133     by (rule countably_compactE)
2134   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E"
2135     and s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"
2136     by (metis (lifting) finite_subset_image C_def)
2137   from s \<open>t \<subseteq> s\<close> have "t \<subseteq> \<Union>E"
2138     using interior_subset by blast
2139   moreover have "finite (\<Union>E)"
2140     using E by auto
2141   ultimately show False using \<open>infinite t\<close>
2142     by (auto simp: finite_subset)
2143 qed
2145 lemma countable_acc_point_imp_seq_compact:
2146   fixes s :: "'a::first_countable_topology set"
2147   assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow>
2148     (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
2149   shows "seq_compact s"
2150 proof -
2151   {
2152     fix f :: "nat \<Rightarrow> 'a"
2153     assume f: "\<forall>n. f n \<in> s"
2154     have "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
2155     proof (cases "finite (range f)")
2156       case True
2157       obtain l where "infinite {n. f n = f l}"
2158         using pigeonhole_infinite[OF _ True] by auto
2159       then obtain r :: "nat \<Rightarrow> nat" where "strict_mono  r" and fr: "\<forall>n. f (r n) = f l"
2160         using infinite_enumerate by blast
2161       then have "strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> f l"
2162         by (simp add: fr o_def)
2163       with f show "\<exists>l\<in>s. \<exists>r. strict_mono  r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
2164         by auto
2165     next
2166       case False
2167       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"
2168         by auto
2169       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..
2170       from this(2) have "\<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
2171         using acc_point_range_imp_convergent_subsequence[of l f] by auto
2172       with \<open>l \<in> s\<close> show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" ..
2173     qed
2174   }
2175   then show ?thesis
2176     unfolding seq_compact_def by auto
2177 qed
2179 lemma seq_compact_eq_countably_compact:
2180   fixes U :: "'a :: first_countable_topology set"
2181   shows "seq_compact U \<longleftrightarrow> countably_compact U"
2182   using
2183     countable_acc_point_imp_seq_compact
2184     countably_compact_imp_acc_point
2185     seq_compact_imp_countably_compact
2186   by metis
2188 lemma seq_compact_eq_acc_point:
2189   fixes s :: "'a :: first_countable_topology set"
2190   shows "seq_compact s \<longleftrightarrow>
2191     (\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)))"
2192   using
2193     countable_acc_point_imp_seq_compact[of s]
2194     countably_compact_imp_acc_point[of s]
2195     seq_compact_imp_countably_compact[of s]
2196   by metis
2198 lemma seq_compact_eq_compact:
2199   fixes U :: "'a :: second_countable_topology set"
2200   shows "seq_compact U \<longleftrightarrow> compact U"
2201   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast
2203 proposition Bolzano_Weierstrass_imp_seq_compact:
2204   fixes s :: "'a::{t1_space, first_countable_topology} set"
2205   shows "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"
2206   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)
2209 subsection%unimportant \<open>Cartesian products\<close>
2211 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"
2212   unfolding seq_compact_def
2213   apply clarify
2214   apply (drule_tac x="fst \<circ> f" in spec)
2215   apply (drule mp, simp add: mem_Times_iff)
2216   apply (clarify, rename_tac l1 r1)
2217   apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
2218   apply (drule mp, simp add: mem_Times_iff)
2219   apply (clarify, rename_tac l2 r2)
2220   apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
2221   apply (rule_tac x="r1 \<circ> r2" in exI)
2222   apply (rule conjI, simp add: strict_mono_def)
2223   apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
2224   apply (drule (1) tendsto_Pair) back
2226   done
2228 lemma compact_Times:
2229   assumes "compact s" "compact t"
2230   shows "compact (s \<times> t)"
2231 proof (rule compactI)
2232   fix C
2233   assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"
2234   have "\<forall>x\<in>s. \<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
2235   proof
2236     fix x
2237     assume "x \<in> s"
2238     have "\<forall>y\<in>t. \<exists>a b c. c \<in> C \<and> open a \<and> open b \<and> x \<in> a \<and> y \<in> b \<and> a \<times> b \<subseteq> c" (is "\<forall>y\<in>t. ?P y")
2239     proof
2240       fix y
2241       assume "y \<in> t"
2242       with \<open>x \<in> s\<close> C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto
2243       then show "?P y" by (auto elim!: open_prod_elim)
2244     qed
2245     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"
2246       and c: "\<And>y. y \<in> t \<Longrightarrow> c y \<in> C \<and> open (a y) \<and> open (b y) \<and> x \<in> a y \<and> y \<in> b y \<and> a y \<times> b y \<subseteq> c y"
2247       by metis
2248     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto
2249     with compactE_image[OF \<open>compact t\<close>] obtain D where D: "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"
2250       by metis
2251     moreover from D c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"
2252       by (fastforce simp: subset_eq)
2253     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
2254       using c by (intro exI[of _ "c`D"] exI[of _ "\<Inter>(a`D)"] conjI) (auto intro!: open_INT)
2255   qed
2256   then obtain a d where a: "\<And>x. x\<in>s \<Longrightarrow> open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"
2257     and d: "\<And>x. x \<in> s \<Longrightarrow> d x \<subseteq> C \<and> finite (d x) \<and> a x \<times> t \<subseteq> \<Union>(d x)"
2258     unfolding subset_eq UN_iff by metis
2259   moreover
2260   from compactE_image[OF \<open>compact s\<close> a]
2261   obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"
2262     by auto
2263   moreover
2264   {
2265     from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"
2266       by auto
2267     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>(d x))"
2268       using d \<open>e \<subseteq> s\<close> by (intro UN_mono) auto
2269     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>(d x))" .
2270   }
2271   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"
2272     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp: subset_eq)
2273 qed
2276 lemma tube_lemma:
2277   assumes "compact K"
2278   assumes "open W"
2279   assumes "{x0} \<times> K \<subseteq> W"
2280   shows "\<exists>X0. x0 \<in> X0 \<and> open X0 \<and> X0 \<times> K \<subseteq> W"
2281 proof -
2282   {
2283     fix y assume "y \<in> K"
2284     then have "(x0, y) \<in> W" using assms by auto
2285     with \<open>open W\<close>
2286     have "\<exists>X0 Y. open X0 \<and> open Y \<and> x0 \<in> X0 \<and> y \<in> Y \<and> X0 \<times> Y \<subseteq> W"
2287       by (rule open_prod_elim) blast
2288   }
2289   then obtain X0 Y where
2290     *: "\<forall>y \<in> K. open (X0 y) \<and> open (Y y) \<and> x0 \<in> X0 y \<and> y \<in> Y y \<and> X0 y \<times> Y y \<subseteq> W"
2291     by metis
2292   from * have "\<forall>t\<in>Y ` K. open t" "K \<subseteq> \<Union>(Y ` K)" by auto
2293   with \<open>compact K\<close> obtain CC where CC: "CC \<subseteq> Y ` K" "finite CC" "K \<subseteq> \<Union>CC"
2294     by (meson compactE)
2295   then obtain c where c: "\<And>C. C \<in> CC \<Longrightarrow> c C \<in> K \<and> C = Y (c C)"
2296     by (force intro!: choice)
2297   with * CC show ?thesis
2298     by (force intro!: exI[where x="\<Inter>C\<in>CC. X0 (c C)"]) (* SLOW *)
2299 qed
2301 lemma continuous_on_prod_compactE:
2302   fixes fx::"'a::topological_space \<times> 'b::topological_space \<Rightarrow> 'c::metric_space"
2303     and e::real
2304   assumes cont_fx: "continuous_on (U \<times> C) fx"
2305   assumes "compact C"
2306   assumes [intro]: "x0 \<in> U"
2307   notes [continuous_intros] = continuous_on_compose2[OF cont_fx]
2308   assumes "e > 0"
2309   obtains X0 where "x0 \<in> X0" "open X0"
2310     "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
2311 proof -
2312   define psi where "psi = (\<lambda>(x, t). dist (fx (x, t)) (fx (x0, t)))"
2313   define W0 where "W0 = {(x, t) \<in> U \<times> C. psi (x, t) < e}"
2314   have W0_eq: "W0 = psi -` {..<e} \<inter> U \<times> C"
2315     by (auto simp: vimage_def W0_def)
2316   have "open {..<e}" by simp
2317   have "continuous_on (U \<times> C) psi"
2318     by (auto intro!: continuous_intros simp: psi_def split_beta')
2319   from this[unfolded continuous_on_open_invariant, rule_format, OF \<open>open {..<e}\<close>]
2320   obtain W where W: "open W" "W \<inter> U \<times> C = W0 \<inter> U \<times> C"
2321     unfolding W0_eq by blast
2322   have "{x0} \<times> C \<subseteq> W \<inter> U \<times> C"
2323     unfolding W
2324     by (auto simp: W0_def psi_def \<open>0 < e\<close>)
2325   then have "{x0} \<times> C \<subseteq> W" by blast
2326   from tube_lemma[OF \<open>compact C\<close> \<open>open W\<close> this]
2327   obtain X0 where X0: "x0 \<in> X0" "open X0" "X0 \<times> C \<subseteq> W"
2328     by blast
2330   have "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
2331   proof safe
2332     fix x assume x: "x \<in> X0" "x \<in> U"
2333     fix t assume t: "t \<in> C"
2334     have "dist (fx (x, t)) (fx (x0, t)) = psi (x, t)"
2335       by (auto simp: psi_def)
2336     also
2337     {
2338       have "(x, t) \<in> X0 \<times> C"
2339         using t x
2340         by auto
2341       also note \<open>\<dots> \<subseteq> W\<close>
2342       finally have "(x, t) \<in> W" .
2343       with t x have "(x, t) \<in> W \<inter> U \<times> C"
2344         by blast
2345       also note \<open>W \<inter> U \<times> C = W0 \<inter> U \<times> C\<close>
2346       finally  have "psi (x, t) < e"
2347         by (auto simp: W0_def)
2348     }
2349     finally show "dist (fx (x, t)) (fx (x0, t)) \<le> e" by simp
2350   qed
2351   from X0(1,2) this show ?thesis ..
2352 qed
2355 subsection \<open>Continuity\<close>
2357 lemma continuous_at_imp_continuous_within:
2358   "continuous (at x) f \<Longrightarrow> continuous (at x within s) f"
2359   unfolding continuous_within continuous_at using Lim_at_imp_Lim_at_within by auto
2361 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f \<longlongrightarrow> l) net"
2362   by simp
2364 lemmas continuous_on = continuous_on_def \<comment> \<open>legacy theorem name\<close>
2366 lemma continuous_within_subset:
2367   "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous (at x within t) f"
2368   unfolding continuous_within by(metis tendsto_within_subset)
2370 lemma continuous_on_interior:
2371   "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"
2372   by (metis continuous_on_eq_continuous_at continuous_on_subset interiorE)
2374 lemma continuous_on_eq:
2375   "\<lbrakk>continuous_on s f; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> continuous_on s g"
2376   unfolding continuous_on_def tendsto_def eventually_at_topological
2377   by simp
2379 text \<open>Characterization of various kinds of continuity in terms of sequences.\<close>
2381 lemma continuous_within_sequentiallyI:
2382   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
2383   assumes "\<And>u::nat \<Rightarrow> 'a. u \<longlonglongrightarrow> a \<Longrightarrow> (\<forall>n. u n \<in> s) \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
2384   shows "continuous (at a within s) f"
2385   using assms unfolding continuous_within tendsto_def[where l = "f a"]
2386   by (auto intro!: sequentially_imp_eventually_within)
2388 lemma continuous_within_tendsto_compose:
2389   fixes f::"'a::t2_space \<Rightarrow> 'b::topological_space"
2390   assumes "continuous (at a within s) f"
2391           "eventually (\<lambda>n. x n \<in> s) F"
2392           "(x \<longlongrightarrow> a) F "
2393   shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
2394 proof -
2395   have *: "filterlim x (inf (nhds a) (principal s)) F"
2396     using assms(2) assms(3) unfolding at_within_def filterlim_inf by (auto simp: filterlim_principal eventually_mono)
2397   show ?thesis
2398     by (auto simp: assms(1) continuous_within[symmetric] tendsto_at_within_iff_tendsto_nhds[symmetric] intro!: filterlim_compose[OF _ *])
2399 qed
2401 lemma continuous_within_tendsto_compose':
2402   fixes f::"'a::t2_space \<Rightarrow> 'b::topological_space"
2403   assumes "continuous (at a within s) f"
2404     "\<And>n. x n \<in> s"
2405     "(x \<longlongrightarrow> a) F "
2406   shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
2407   by (auto intro!: continuous_within_tendsto_compose[OF assms(1)] simp add: assms)
2409 lemma continuous_within_sequentially:
2410   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
2411   shows "continuous (at a within s) f \<longleftrightarrow>
2412     (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
2413          \<longrightarrow> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
2414   using continuous_within_tendsto_compose'[of a s f _ sequentially]
2415     continuous_within_sequentiallyI[of a s f]
2416   by (auto simp: o_def)
2418 lemma continuous_at_sequentiallyI:
2419   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
2420   assumes "\<And>u. u \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
2421   shows "continuous (at a) f"
2422   using continuous_within_sequentiallyI[of a UNIV f] assms by auto
2424 lemma continuous_at_sequentially:
2425   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
2426   shows "continuous (at a) f \<longleftrightarrow>
2427     (\<forall>x. (x \<longlongrightarrow> a) sequentially --> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
2428   using continuous_within_sequentially[of a UNIV f] by simp
2430 lemma continuous_on_sequentiallyI:
2431   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
2432   assumes "\<And>u a. (\<forall>n. u n \<in> s) \<Longrightarrow> a \<in> s \<Longrightarrow> u \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
2433   shows "continuous_on s f"
2434   using assms unfolding continuous_on_eq_continuous_within
2435   using continuous_within_sequentiallyI[of _ s f] by auto
2437 lemma continuous_on_sequentially:
2438   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
2439   shows "continuous_on s f \<longleftrightarrow>
2440     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
2441       --> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
2442     (is "?lhs = ?rhs")
2443 proof
2444   assume ?rhs
2445   then show ?lhs
2446     using continuous_within_sequentially[of _ s f]
2447     unfolding continuous_on_eq_continuous_within
2448     by auto
2449 next
2450   assume ?lhs
2451   then show ?rhs
2452     unfolding continuous_on_eq_continuous_within
2453     using continuous_within_sequentially[of _ s f]
2454     by auto
2455 qed
2457 text \<open>Continuity in terms of open preimages.\<close>
2459 lemma continuous_at_open:
2460   "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"
2461   unfolding continuous_within_topological [of x UNIV f]
2462   unfolding imp_conjL
2463   by (intro all_cong imp_cong ex_cong conj_cong refl) auto
2465 lemma continuous_imp_tendsto:
2466   assumes "continuous (at x0) f"
2467     and "x \<longlonglongrightarrow> x0"
2468   shows "(f \<circ> x) \<longlonglongrightarrow> (f x0)"
2469 proof (rule topological_tendstoI)
2470   fix S
2471   assume "open S" "f x0 \<in> S"
2472   then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"
2473      using assms continuous_at_open by metis
2474   then have "eventually (\<lambda>n. x n \<in> T) sequentially"
2475     using assms T_def by (auto simp: tendsto_def)
2476   then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"
2477     using T_def by (auto elim!: eventually_mono)
2478 qed
2480 subsection \<open>Homeomorphisms\<close>
2482 definition%important "homeomorphism s t f g \<longleftrightarrow>
2483   (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
2484   (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
2486 lemma homeomorphismI [intro?]:
2487   assumes "continuous_on S f" "continuous_on T g"
2488           "f ` S \<subseteq> T" "g ` T \<subseteq> S" "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>y. y \<in> T \<Longrightarrow> f(g y) = y"
2489     shows "homeomorphism S T f g"
2490   using assms by (force simp: homeomorphism_def)
2492 lemma homeomorphism_translation:
2493   fixes a :: "'a :: real_normed_vector"
2494   shows "homeomorphism ((+) a ` S) S ((+) (- a)) ((+) a)"
2495 unfolding homeomorphism_def by (auto simp: algebra_simps continuous_intros)
2497 lemma homeomorphism_ident: "homeomorphism T T (\<lambda>a. a) (\<lambda>a. a)"
2498   by (rule homeomorphismI) auto
2500 lemma homeomorphism_compose:
2501   assumes "homeomorphism S T f g" "homeomorphism T U h k"
2502     shows "homeomorphism S U (h o f) (g o k)"
2503   using assms
2504   unfolding homeomorphism_def
2505   by (intro conjI ballI continuous_on_compose) (auto simp: image_iff)
2508 lemma homeomorphism_symD: "homeomorphism S t f g \<Longrightarrow> homeomorphism t S g f"
2511 lemma homeomorphism_sym: "homeomorphism S t f g = homeomorphism t S g f"
2512   by (force simp: homeomorphism_def)
2514 definition%important homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
2515     (infixr "homeomorphic" 60)
2516   where "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
2518 lemma homeomorphic_empty [iff]:
2519      "S homeomorphic {} \<longleftrightarrow> S = {}" "{} homeomorphic S \<longleftrightarrow> S = {}"
2520   by (auto simp: homeomorphic_def homeomorphism_def)
2522 lemma homeomorphic_refl: "s homeomorphic s"
2523   unfolding homeomorphic_def homeomorphism_def
2524   using continuous_on_id
2525   apply (rule_tac x = "(\<lambda>x. x)" in exI)
2526   apply (rule_tac x = "(\<lambda>x. x)" in exI)
2527   apply blast
2528   done
2530 lemma homeomorphic_sym: "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
2531   unfolding homeomorphic_def homeomorphism_def
2532   by blast
2534 lemma homeomorphic_trans [trans]:
2535   assumes "S homeomorphic T"
2536       and "T homeomorphic U"
2537     shows "S homeomorphic U"
2538   using assms
2539   unfolding homeomorphic_def
2540 by (metis homeomorphism_compose)
2542 lemma homeomorphic_minimal:
2543   "s homeomorphic t \<longleftrightarrow>
2544     (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
2545            (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
2546            continuous_on s f \<and> continuous_on t g)"
2547    (is "?lhs = ?rhs")
2548 proof
2549   assume ?lhs
2550   then show ?rhs
2551     by (fastforce simp: homeomorphic_def homeomorphism_def)
2552 next
2553   assume ?rhs
2554   then show ?lhs
2555     apply clarify
2556     unfolding homeomorphic_def homeomorphism_def
2557     by (metis equalityI image_subset_iff subsetI)
2558  qed
2560 lemma homeomorphicI [intro?]:
2561    "\<lbrakk>f ` S = T; g ` T = S;
2562      continuous_on S f; continuous_on T g;
2563      \<And>x. x \<in> S \<Longrightarrow> g(f(x)) = x;
2564      \<And>y. y \<in> T \<Longrightarrow> f(g(y)) = y\<rbrakk> \<Longrightarrow> S homeomorphic T"
2565 unfolding homeomorphic_def homeomorphism_def by metis
2567 lemma homeomorphism_of_subsets:
2568    "\<lbrakk>homeomorphism S T f g; S' \<subseteq> S; T'' \<subseteq> T; f ` S' = T'\<rbrakk>
2569     \<Longrightarrow> homeomorphism S' T' f g"
2570 apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
2571 by (metis subsetD imageI)
2573 lemma homeomorphism_apply1: "\<lbrakk>homeomorphism S T f g; x \<in> S\<rbrakk> \<Longrightarrow> g(f x) = x"
2576 lemma homeomorphism_apply2: "\<lbrakk>homeomorphism S T f g; x \<in> T\<rbrakk> \<Longrightarrow> f(g x) = x"
2579 lemma homeomorphism_image1: "homeomorphism S T f g \<Longrightarrow> f ` S = T"
2582 lemma homeomorphism_image2: "homeomorphism S T f g \<Longrightarrow> g ` T = S"
2585 lemma homeomorphism_cont1: "homeomorphism S T f g \<Longrightarrow> continuous_on S f"
2588 lemma homeomorphism_cont2: "homeomorphism S T f g \<Longrightarrow> continuous_on T g"
2591 lemma continuous_on_no_limpt:
2592    "(\<And>x. \<not> x islimpt S) \<Longrightarrow> continuous_on S f"
2593   unfolding continuous_on_def
2594   by (metis UNIV_I empty_iff eventually_at_topological islimptE open_UNIV tendsto_def trivial_limit_within)
2596 lemma continuous_on_finite:
2597   fixes S :: "'a::t1_space set"
2598   shows "finite S \<Longrightarrow> continuous_on S f"
2599 by (metis continuous_on_no_limpt islimpt_finite)
2601 lemma homeomorphic_finite:
2602   fixes S :: "'a::t1_space set" and T :: "'b::t1_space set"
2603   assumes "finite T"
2604   shows "S homeomorphic T \<longleftrightarrow> finite S \<and> finite T \<and> card S = card T" (is "?lhs = ?rhs")
2605 proof
2606   assume "S homeomorphic T"
2607   with assms show ?rhs
2608     apply (auto simp: homeomorphic_def homeomorphism_def)
2609      apply (metis finite_imageI)
2610     by (metis card_image_le finite_imageI le_antisym)
2611 next
2612   assume R: ?rhs
2613   with finite_same_card_bij obtain h where "bij_betw h S T"
2614     by auto
2615   with R show ?lhs
2616     apply (auto simp: homeomorphic_def homeomorphism_def continuous_on_finite)
2617     apply (rule_tac x=h in exI)
2618     apply (rule_tac x="inv_into S h" in exI)
2619     apply (auto simp:  bij_betw_inv_into_left bij_betw_inv_into_right bij_betw_imp_surj_on inv_into_into bij_betwE)
2620     apply (metis bij_betw_def bij_betw_inv_into)
2621     done
2622 qed
2624 text \<open>Relatively weak hypotheses if a set is compact.\<close>
2626 lemma homeomorphism_compact:
2627   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
2628   assumes "compact s" "continuous_on s f"  "f ` s = t"  "inj_on f s"
2629   shows "\<exists>g. homeomorphism s t f g"
2630 proof -
2631   define g where "g x = (SOME y. y\<in>s \<and> f y = x)" for x
2632   have g: "\<forall>x\<in>s. g (f x) = x"
2633     using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
2634   {
2635     fix y
2636     assume "y \<in> t"
2637     then obtain x where x:"f x = y" "x\<in>s"
2638       using assms(3) by auto
2639     then have "g (f x) = x" using g by auto
2640     then have "f (g y) = y" unfolding x(1)[symmetric] by auto
2641   }
2642   then have g':"\<forall>x\<in>t. f (g x) = x" by auto
2643   moreover
2644   {
2645     fix x
2646     have "x\<in>s \<Longrightarrow> x \<in> g ` t"
2647       using g[THEN bspec[where x=x]]
2648       unfolding image_iff
2649       using assms(3)
2650       by (auto intro!: bexI[where x="f x"])
2651     moreover
2652     {
2653       assume "x\<in>g ` t"
2654       then obtain y where y:"y\<in>t" "g y = x" by auto
2655       then obtain x' where x':"x'\<in>s" "f x' = y"
2656         using assms(3) by auto
2657       then have "x \<in> s"
2658         unfolding g_def
2659         using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"]
2660         unfolding y(2)[symmetric] and g_def
2661         by auto
2662     }
2663     ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" ..
2664   }
2665   then have "g ` t = s" by auto
2666   ultimately show ?thesis
2667     unfolding homeomorphism_def homeomorphic_def
2668     apply (rule_tac x=g in exI)
2669     using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2)
2670     apply auto
2671     done
2672 qed
2674 lemma homeomorphic_compact:
2675   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
2676   shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s \<Longrightarrow> s homeomorphic t"
2677   unfolding homeomorphic_def by (metis homeomorphism_compact)
2679 text\<open>Preservation of topological properties.\<close>
2681 lemma homeomorphic_compactness: "s homeomorphic t \<Longrightarrow> (compact s \<longleftrightarrow> compact t)"
2682   unfolding homeomorphic_def homeomorphism_def
2683   by (metis compact_continuous_image)
2686 subsection%unimportant \<open>On Linorder Topologies\<close>
2688 lemma islimpt_greaterThanLessThan1:
2689   fixes a b::"'a::{linorder_topology, dense_order}"
2690   assumes "a < b"
2691   shows  "a islimpt {a<..<b}"
2692 proof (rule islimptI)
2693   fix T
2694   assume "open T" "a \<in> T"
2695   from open_right[OF this \<open>a < b\<close>]
2696   obtain c where c: "a < c" "{a..<c} \<subseteq> T" by auto
2697   with assms dense[of a "min c b"]
2698   show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> a"
2699     by (metis atLeastLessThan_iff greaterThanLessThan_iff min_less_iff_conj
2700       not_le order.strict_implies_order subset_eq)
2701 qed
2703 lemma islimpt_greaterThanLessThan2:
2704   fixes a b::"'a::{linorder_topology, dense_order}"
2705   assumes "a < b"
2706   shows  "b islimpt {a<..<b}"
2707 proof (rule islimptI)
2708   fix T
2709   assume "open T" "b \<in> T"
2710   from open_left[OF this \<open>a < b\<close>]
2711   obtain c where c: "c < b" "{c<..b} \<subseteq> T" by auto
2712   with assms dense[of "max a c" b]
2713   show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> b"
2714     by (metis greaterThanAtMost_iff greaterThanLessThan_iff max_less_iff_conj
2715       not_le order.strict_implies_order subset_eq)
2716 qed
2718 lemma closure_greaterThanLessThan[simp]:
2719   fixes a b::"'a::{linorder_topology, dense_order}"
2720   shows "a < b \<Longrightarrow> closure {a <..< b} = {a .. b}" (is "_ \<Longrightarrow> ?l = ?r")
2721 proof
2722   have "?l \<subseteq> closure ?r"
2723     by (rule closure_mono) auto
2724   thus "closure {a<..<b} \<subseteq> {a..b}" by simp
2725 qed (auto simp: closure_def order.order_iff_strict islimpt_greaterThanLessThan1
2726   islimpt_greaterThanLessThan2)
2728 lemma closure_greaterThan[simp]:
2729   fixes a b::"'a::{no_top, linorder_topology, dense_order}"
2730   shows "closure {a<..} = {a..}"
2731 proof -
2732   from gt_ex obtain b where "a < b" by auto
2733   hence "{a<..} = {a<..<b} \<union> {b..}" by auto
2734   also have "closure \<dots> = {a..}" using \<open>a < b\<close> unfolding closure_Un
2735     by auto
2736   finally show ?thesis .
2737 qed
2739 lemma closure_lessThan[simp]:
2740   fixes b::"'a::{no_bot, linorder_topology, dense_order}"
2741   shows "closure {..<b} = {..b}"
2742 proof -
2743   from lt_ex obtain a where "a < b" by auto
2744   hence "{..<b} = {a<..<b} \<union> {..a}" by auto
2745   also have "closure \<dots> = {..b}" using \<open>a < b\<close> unfolding closure_Un
2746     by auto
2747   finally show ?thesis .
2748 qed
2750 lemma closure_atLeastLessThan[simp]:
2751   fixes a b::"'a::{linorder_topology, dense_order}"
2752   assumes "a < b"
2753   shows "closure {a ..< b} = {a .. b}"
2754 proof -
2755   from assms have "{a ..< b} = {a} \<union> {a <..< b}" by auto
2756   also have "closure \<dots> = {a .. b}" unfolding closure_Un
2757     by (auto simp: assms less_imp_le)
2758   finally show ?thesis .
2759 qed
2761 lemma closure_greaterThanAtMost[simp]:
2762   fixes a b::"'a::{linorder_topology, dense_order}"
2763   assumes "a < b"
2764   shows "closure {a <.. b} = {a .. b}"
2765 proof -
2766   from assms have "{a <.. b} = {b} \<union> {a <..< b}" by auto
2767   also have "closure \<dots> = {a .. b}" unfolding closure_Un
2768     by (auto simp: assms less_imp_le)
2769   finally show ?thesis .
2770 qed
2773 end