src/HOL/Analysis/Homeomorphism.thy
changeset 69739 8b47c021666e
parent 69722 b5163b2132c5
child 69922 4a9167f377b0
equal deleted inserted replaced
69738:c558fef62915 69739:8b47c021666e
   276       qed
   276       qed
   277     qed
   277     qed
   278   qed
   278   qed
   279 qed
   279 qed
   280 
   280 
   281 proposition (*FIXME needs name *)
   281 proposition
   282   fixes S :: "'a::euclidean_space set"
   282   fixes S :: "'a::euclidean_space set"
   283   assumes "compact S" and 0: "0 \<in> rel_interior S"
   283   assumes "compact S" and 0: "0 \<in> rel_interior S"
   284       and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment 0 x \<subseteq> rel_interior S"
   284       and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment 0 x \<subseteq> rel_interior S"
   285     shows starlike_compact_projective1_0:
   285     shows starlike_compact_projective1_0:
   286             "S - rel_interior S homeomorphic sphere 0 1 \<inter> affine hull S"
   286             "S - rel_interior S homeomorphic sphere 0 1 \<inter> affine hull S"
   538     apply (subst homeomorphic_sym)
   538     apply (subst homeomorphic_sym)
   539     apply (rule homeomorphic_compact [OF co01 cont_nosp2 [unfolded o_def] im_cball inj_cball])
   539     apply (rule homeomorphic_compact [OF co01 cont_nosp2 [unfolded o_def] im_cball inj_cball])
   540     done
   540     done
   541 qed
   541 qed
   542 
   542 
   543 corollary (* FIXME needs name *)
   543 corollary
   544   fixes S :: "'a::euclidean_space set"
   544   fixes S :: "'a::euclidean_space set"
   545   assumes "compact S" and a: "a \<in> rel_interior S"
   545   assumes "compact S" and a: "a \<in> rel_interior S"
   546       and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
   546       and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
   547     shows starlike_compact_projective1:
   547     shows starlike_compact_projective1:
   548             "S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S"
   548             "S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S"