(* Title: HOL/Analysis/Homeomorphism.thy Author: LC Paulson (ported from HOL Light) *) section ‹Homeomorphism Theorems› theory Homeomorphism imports Homotopy begin lemma homeomorphic_spheres': fixes a ::"'a::euclidean_space" and b ::"'b::euclidean_space" assumes "0 < δ" and dimeq: "DIM('a) = DIM('b)" shows "(sphere a δ) homeomorphic (sphere b δ)" proof - obtain f :: "'a⇒'b" and g where "linear f" "linear g" and fg: "⋀x. norm(f x) = norm x" "⋀y. norm(g y) = norm y" "⋀x. g(f x) = x" "⋀y. f(g y) = y" by (blast intro: isomorphisms_UNIV_UNIV [OF dimeq]) then have "continuous_on UNIV f" "continuous_on UNIV g" using linear_continuous_on linear_linear by blast+ then show ?thesis unfolding homeomorphic_minimal apply(rule_tac x="λx. b + f(x - a)" in exI) apply(rule_tac x="λx. a + g(x - b)" in exI) using assms apply (force intro: continuous_intros continuous_on_compose2 [of _ f] continuous_on_compose2 [of _ g] simp: dist_commute dist_norm fg) done qed lemma homeomorphic_spheres_gen: fixes a :: "'a::euclidean_space" and b :: "'b::euclidean_space" assumes "0 < r" "0 < s" "DIM('a::euclidean_space) = DIM('b::euclidean_space)" shows "(sphere a r homeomorphic sphere b s)" using assms homeomorphic_trans [OF homeomorphic_spheres homeomorphic_spheres'] by auto subsection ‹Homeomorphism of all convex compact sets with nonempty interior› proposition fixes S :: "'a::euclidean_space set" assumes "compact S" and 0: "0 ∈ rel_interior S" and star: "⋀x. x ∈ S ⟹ open_segment 0 x ⊆ rel_interior S" shows starlike_compact_projective1_0: "S - rel_interior S homeomorphic sphere 0 1 ∩ affine hull S" (is "?SMINUS homeomorphic ?SPHER") and starlike_compact_projective2_0: "S homeomorphic cball 0 1 ∩ affine hull S" (is "S homeomorphic ?CBALL") proof - have starI: "(u *⇩_{R}x) ∈ rel_interior S" if "x ∈ S" "0 ≤ u" "u < 1" for x u proof (cases "x=0 ∨ u=0") case True with 0 show ?thesis by force next case False with that show ?thesis by (auto simp: in_segment intro: star [THEN subsetD]) qed have "0 ∈ S" using assms rel_interior_subset by auto define proj where "proj ≡ λx::'a. x /⇩_{R}norm x" have eqI: "x = y" if "proj x = proj y" "norm x = norm y" for x y using that by (force simp: proj_def) then have iff_eq: "⋀x y. (proj x = proj y ∧ norm x = norm y) ⟷ x = y" by blast have projI: "x ∈ affine hull S ⟹ proj x ∈ affine hull S" for x by (metis ‹0 ∈ S› affine_hull_span_0 hull_inc span_mul proj_def) have nproj1 [simp]: "x ≠ 0 ⟹ norm(proj x) = 1" for x by (simp add: proj_def) have proj0_iff [simp]: "proj x = 0 ⟷ x = 0" for x by (simp add: proj_def) have cont_proj: "continuous_on (UNIV - {0}) proj" unfolding proj_def by (rule continuous_intros | force)+ have proj_spherI: "⋀x. ⟦x ∈ affine hull S; x ≠ 0⟧ ⟹ proj x ∈ ?SPHER" by (simp add: projI) have "bounded S" "closed S" using ‹compact S› compact_eq_bounded_closed by blast+ have inj_on_proj: "inj_on proj (S - rel_interior S)" proof fix x y assume x: "x ∈ S - rel_interior S" and y: "y ∈ S - rel_interior S" and eq: "proj x = proj y" then have xynot: "x ≠ 0" "y ≠ 0" "x ∈ S" "y ∈ S" "x ∉ rel_interior S" "y ∉ rel_interior S" using 0 by auto consider "norm x = norm y" | "norm x < norm y" | "norm x > norm y" by linarith then show "x = y" proof cases assume "norm x = norm y" with iff_eq eq show "x = y" by blast next assume *: "norm x < norm y" have "x /⇩_{R}norm x = norm x *⇩_{R}(x /⇩_{R}norm x) /⇩_{R}norm (norm x *⇩_{R}(x /⇩_{R}norm x))" by force then have "proj ((norm x / norm y) *⇩_{R}y) = proj x" by (metis (no_types) divide_inverse local.proj_def eq scaleR_scaleR) then have [simp]: "(norm x / norm y) *⇩_{R}y = x" by (rule eqI) (simp add: ‹y ≠ 0›) have no: "0 ≤ norm x / norm y" "norm x / norm y < 1" using * by (auto simp: field_split_simps) then show "x = y" using starI [OF ‹y ∈ S› no] xynot by auto next assume *: "norm x > norm y" have "y /⇩_{R}norm y = norm y *⇩_{R}(y /⇩_{R}norm y) /⇩_{R}norm (norm y *⇩_{R}(y /⇩_{R}norm y))" by force then have "proj ((norm y / norm x) *⇩_{R}x) = proj y" by (metis (no_types) divide_inverse local.proj_def eq scaleR_scaleR) then have [simp]: "(norm y / norm x) *⇩_{R}x = y" by (rule eqI) (simp add: ‹x ≠ 0›) have no: "0 ≤ norm y / norm x" "norm y / norm x < 1" using * by (auto simp: field_split_simps) then show "x = y" using starI [OF ‹x ∈ S› no] xynot by auto qed qed have "∃surf. homeomorphism (S - rel_interior S) ?SPHER proj surf" proof (rule homeomorphism_compact) show "compact (S - rel_interior S)" using ‹compact S› compact_rel_boundary by blast show "continuous_on (S - rel_interior S) proj" using 0 by (blast intro: continuous_on_subset [OF cont_proj]) show "proj ` (S - rel_interior S) = ?SPHER" proof show "proj ` (S - rel_interior S) ⊆ ?SPHER" using 0 by (force simp: hull_inc projI intro: nproj1) show "?SPHER ⊆ proj ` (S - rel_interior S)" proof (clarsimp simp: proj_def) fix x assume "x ∈ affine hull S" and nox: "norm x = 1" then have "x ≠ 0" by auto obtain d where "0 < d" and dx: "(d *⇩_{R}x) ∈ rel_frontier S" and ri: "⋀e. ⟦0 ≤ e; e < d⟧ ⟹ (e *⇩_{R}x) ∈ rel_interior S" using ray_to_rel_frontier [OF ‹bounded S› 0] ‹x ∈ affine hull S› ‹x ≠ 0› by auto show "x ∈ (λx. x /⇩_{R}norm x) ` (S - rel_interior S)" proof show "x = d *⇩_{R}x /⇩_{R}norm (d *⇩_{R}x)" using ‹0 < d› by (auto simp: nox) show "d *⇩_{R}x ∈ S - rel_interior S" using dx ‹closed S› by (auto simp: rel_frontier_def) qed qed qed qed (rule inj_on_proj) then obtain surf where surf: "homeomorphism (S - rel_interior S) ?SPHER proj surf" by blast then have cont_surf: "continuous_on (proj ` (S - rel_interior S)) surf" by (auto simp: homeomorphism_def) have surf_nz: "⋀x. x ∈ ?SPHER ⟹ surf x ≠ 0" by (metis "0" DiffE homeomorphism_def imageI surf) have cont_nosp: "continuous_on (?SPHER) (λx. norm x *⇩_{R}((surf o proj) x))" proof (intro continuous_intros) show "continuous_on (sphere 0 1 ∩ affine hull S) proj" by (rule continuous_on_subset [OF cont_proj], force) show "continuous_on (proj ` (sphere 0 1 ∩ affine hull S)) surf" by (intro continuous_on_subset [OF cont_surf]) (force simp: homeomorphism_image1 [OF surf] dest: proj_spherI) qed have surfpS: "⋀x. ⟦norm x = 1; x ∈ affine hull S⟧ ⟹ surf (proj x) ∈ S" by (metis (full_types) DiffE ‹0 ∈ S› homeomorphism_def image_eqI norm_zero proj_spherI real_vector.scale_zero_left scaleR_one surf) have *: "∃y. norm y = 1 ∧ y ∈ affine hull S ∧ x = surf (proj y)" if "x ∈ S" "x ∉ rel_interior S" for x proof - have "proj x ∈ ?SPHER" by (metis (full_types) "0" hull_inc proj_spherI that) moreover have "surf (proj x) = x" by (metis Diff_iff homeomorphism_def surf that) ultimately show ?thesis by (metis ‹⋀x. x ∈ ?SPHER ⟹ surf x ≠ 0› hull_inc inverse_1 local.proj_def norm_sgn projI scaleR_one sgn_div_norm that(1)) qed have surfp_notin: "⋀x. ⟦norm x = 1; x ∈ affine hull S⟧ ⟹ surf (proj x) ∉ rel_interior S" by (metis (full_types) DiffE one_neq_zero homeomorphism_def image_eqI norm_zero proj_spherI surf) have no_sp_im: "(λx. norm x *⇩_{R}surf (proj x)) ` (?SPHER) = S - rel_interior S" by (auto simp: surfpS image_def Bex_def surfp_notin *) have inj_spher: "inj_on (λx. norm x *⇩_{R}surf (proj x)) ?SPHER" proof fix x y assume xy: "x ∈ ?SPHER" "y ∈ ?SPHER" and eq: " norm x *⇩_{R}surf (proj x) = norm y *⇩_{R}surf (proj y)" then have "norm x = 1" "norm y = 1" "x ∈ affine hull S" "y ∈ affine hull S" using 0 by auto with eq show "x = y" by (simp add: proj_def) (metis surf xy homeomorphism_def) qed have co01: "compact ?SPHER" by (simp add: compact_Int_closed) show "?SMINUS homeomorphic ?SPHER" using homeomorphic_def surf by blast have proj_scaleR: "⋀a x. 0 < a ⟹ proj (a *⇩_{R}x) = proj x" by (simp add: proj_def) have cont_sp0: "continuous_on (affine hull S - {0}) (surf o proj)" proof (rule continuous_on_compose [OF continuous_on_subset [OF cont_proj]]) show "continuous_on (proj ` (affine hull S - {0})) surf" using homeomorphism_image1 proj_spherI surf by (intro continuous_on_subset [OF cont_surf]) fastforce qed auto obtain B where "B>0" and B: "⋀x. x ∈ S ⟹ norm x ≤ B" by (metis compact_imp_bounded ‹compact S› bounded_pos_less less_eq_real_def) have cont_nosp: "continuous (at x within ?CBALL) (λx. norm x *⇩_{R}surf (proj x))" if "norm x ≤ 1" "x ∈ affine hull S" for x proof (cases "x=0") case True have "(norm ⤏ 0) (at 0 within cball 0 1 ∩ affine hull S)" by (simp add: tendsto_norm_zero eventually_at) with True show ?thesis apply (simp add: continuous_within) apply (rule lim_null_scaleR_bounded [where B=B]) using B ‹0 < B› local.proj_def projI surfpS by (auto simp: eventually_at) next case False then have "∀⇩_{F}x in at x. (x ∈ affine hull S - {0}) = (x ∈ affine hull S)" by (force simp: False eventually_at) moreover have "continuous (at x within affine hull S - {0}) (λx. surf (proj x))" using cont_sp0 False that by (auto simp add: continuous_on_eq_continuous_within) ultimately have *: "continuous (at x within affine hull S) (λx. surf (proj x))" by (simp add: continuous_within Lim_transform_within_set continuous_on_eq_continuous_within) show ?thesis by (intro continuous_within_subset [where S = "affine hull S", OF _ Int_lower2] continuous_intros *) qed have cont_nosp2: "continuous_on ?CBALL (λx. norm x *⇩_{R}((surf o proj) x))" by (simp add: continuous_on_eq_continuous_within cont_nosp) have "norm y *⇩_{R}surf (proj y) ∈ S" if "y ∈ cball 0 1" and yaff: "y ∈ affine hull S" for y proof (cases "y=0") case True then show ?thesis by (simp add: ‹0 ∈ S›) next case False then have "norm y *⇩_{R}surf (proj y) = norm y *⇩_{R}surf (proj (y /⇩_{R}norm y))" by (simp add: proj_def) have "norm y ≤ 1" using that by simp have "surf (proj (y /⇩_{R}norm y)) ∈ S" using False local.proj_def nproj1 projI surfpS yaff by blast then have "surf (proj y) ∈ S" by (simp add: False proj_def) then show "norm y *⇩_{R}surf (proj y) ∈ S" by (metis dual_order.antisym le_less_linear norm_ge_zero rel_interior_subset scaleR_one starI subset_eq ‹norm y ≤ 1›) qed moreover have "x ∈ (λx. norm x *⇩_{R}surf (proj x)) ` (?CBALL)" if "x ∈ S" for x proof (cases "x=0") case True with that hull_inc show ?thesis by fastforce next case False then have psp: "proj (surf (proj x)) = proj x" by (metis homeomorphism_def hull_inc proj_spherI surf that) have nxx: "norm x *⇩_{R}proj x = x" by (simp add: False local.proj_def) have affineI: "(1 / norm (surf (proj x))) *⇩_{R}x ∈ affine hull S" by (metis ‹0 ∈ S› affine_hull_span_0 hull_inc span_clauses(4) that) have sproj_nz: "surf (proj x) ≠ 0" by (metis False proj0_iff psp) then have "proj x = proj (proj x)" by (metis False nxx proj_scaleR zero_less_norm_iff) moreover have scaleproj: "⋀a r. r *⇩_{R}proj a = (r / norm a) *⇩_{R}a" by (simp add: divide_inverse local.proj_def) ultimately have "(norm (surf (proj x)) / norm x) *⇩_{R}x ∉ rel_interior S" by (metis (no_types) sproj_nz divide_self_if hull_inc norm_eq_zero nproj1 projI psp scaleR_one surfp_notin that) then have "(norm (surf (proj x)) / norm x) ≥ 1" using starI [OF that] by (meson starI [OF that] le_less_linear norm_ge_zero zero_le_divide_iff) then have nole: "norm x ≤ norm (surf (proj x))" by (simp add: le_divide_eq_1) let ?inx = "x /⇩_{R}norm (surf (proj x))" show ?thesis proof show "x = norm ?inx *⇩_{R}surf (proj ?inx)" by (simp add: field_simps) (metis inverse_eq_divide nxx positive_imp_inverse_positive proj_scaleR psp scaleproj sproj_nz zero_less_norm_iff) qed (auto simp: field_split_simps nole affineI) qed ultimately have im_cball: "(λx. norm x *⇩_{R}surf (proj x)) ` ?CBALL = S" by blast have inj_cball: "inj_on (λx. norm x *⇩_{R}surf (proj x)) ?CBALL" proof fix x y assume "x ∈ ?CBALL" "y ∈ ?CBALL" and eq: "norm x *⇩_{R}surf (proj x) = norm y *⇩_{R}surf (proj y)" then have x: "x ∈ affine hull S" and y: "y ∈ affine hull S" using 0 by auto show "x = y" proof (cases "x=0 ∨ y=0") case True then show "x = y" using eq proj_spherI surf_nz x y by force next case False with x y have speq: "surf (proj x) = surf (proj y)" by (metis eq homeomorphism_apply2 proj_scaleR proj_spherI surf zero_less_norm_iff) then have "norm x = norm y" by (metis ‹x ∈ affine hull S› ‹y ∈ affine hull S› eq proj_spherI real_vector.scale_cancel_right surf_nz) moreover have "proj x = proj y" by (metis (no_types) False speq homeomorphism_apply2 proj_spherI surf x y) ultimately show "x = y" using eq eqI by blast qed qed have co01: "compact ?CBALL" by (simp add: compact_Int_closed) show "S homeomorphic ?CBALL" using homeomorphic_compact [OF co01 cont_nosp2 [unfolded o_def] im_cball inj_cball] homeomorphic_sym by blast qed corollary fixes S :: "'a::euclidean_space set" assumes "compact S" and a: "a ∈ rel_interior S" and star: "⋀x. x ∈ S ⟹ open_segment a x ⊆ rel_interior S" shows starlike_compact_projective1: "S - rel_interior S homeomorphic sphere a 1 ∩ affine hull S" and starlike_compact_projective2: "S homeomorphic cball a 1 ∩ affine hull S" proof - have 1: "compact ((+) (-a) ` S)" by (meson assms compact_translation) have 2: "0 ∈ rel_interior ((+) (-a) ` S)" using a rel_interior_translation [of "- a" S] by (simp cong: image_cong_simp) have 3: "open_segment 0 x ⊆ rel_interior ((+) (-a) ` S)" if "x ∈ ((+) (-a) ` S)" for x proof - have "x+a ∈ S" using that by auto then have "open_segment a (x+a) ⊆ rel_interior S" by (metis star) then show ?thesis using open_segment_translation [of a 0 x] using rel_interior_translation [of "- a" S] by (fastforce simp add: ac_simps image_iff cong: image_cong_simp) qed have "S - rel_interior S homeomorphic ((+) (-a) ` S) - rel_interior ((+) (-a) ` S)" by (metis rel_interior_translation translation_diff homeomorphic_translation) also have "... homeomorphic sphere 0 1 ∩ affine hull ((+) (-a) ` S)" by (rule starlike_compact_projective1_0 [OF 1 2 3]) also have "... = (+) (-a) ` (sphere a 1 ∩ affine hull S)" by (metis affine_hull_translation left_minus sphere_translation translation_Int) also have "... homeomorphic sphere a 1 ∩ affine hull S" using homeomorphic_translation homeomorphic_sym by blast finally show "S - rel_interior S homeomorphic sphere a 1 ∩ affine hull S" . have "S homeomorphic ((+) (-a) ` S)" by (metis homeomorphic_translation) also have "... homeomorphic cball 0 1 ∩ affine hull ((+) (-a) ` S)" by (rule starlike_compact_projective2_0 [OF 1 2 3]) also have "... = (+) (-a) ` (cball a 1 ∩ affine hull S)" by (metis affine_hull_translation left_minus cball_translation translation_Int) also have "... homeomorphic cball a 1 ∩ affine hull S" using homeomorphic_translation homeomorphic_sym by blast finally show "S homeomorphic cball a 1 ∩ affine hull S" . qed corollary starlike_compact_projective_special: assumes "compact S" and cb01: "cball (0::'a::euclidean_space) 1 ⊆ S" and scale: "⋀x u. ⟦x ∈ S; 0 ≤ u; u < 1⟧ ⟹ u *⇩_{R}x ∈ S - frontier S" shows "S homeomorphic (cball (0::'a::euclidean_space) 1)" proof - have "ball 0 1 ⊆ interior S" using cb01 interior_cball interior_mono by blast then have 0: "0 ∈ rel_interior S" by (meson centre_in_ball subsetD interior_subset_rel_interior le_numeral_extra(2) not_le) have [simp]: "affine hull S = UNIV" using ‹ball 0 1 ⊆ interior S› by (auto intro!: affine_hull_nonempty_interior) have star: "open_segment 0 x ⊆ rel_interior S" if "x ∈ S" for x proof fix p assume "p ∈ open_segment 0 x" then obtain u where "x ≠ 0" and u: "0 ≤ u" "u < 1" and p: "u *⇩_{R}x = p" by (auto simp: in_segment) then show "p ∈ rel_interior S" using scale [OF that u] closure_subset frontier_def interior_subset_rel_interior by fastforce qed show ?thesis using starlike_compact_projective2_0 [OF ‹compact S› 0 star] by simp qed lemma homeomorphic_convex_lemma: fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" assumes "convex S" "compact S" "convex T" "compact T" and affeq: "aff_dim S = aff_dim T" shows "(S - rel_interior S) homeomorphic (T - rel_interior T) ∧ S homeomorphic T" proof (cases "rel_interior S = {} ∨ rel_interior T = {}") case True then show ?thesis by (metis Diff_empty affeq ‹convex S› ‹convex T› aff_dim_empty homeomorphic_empty rel_interior_eq_empty aff_dim_empty) next case False then obtain a b where a: "a ∈ rel_interior S" and b: "b ∈ rel_interior T" by auto have starS: "⋀x. x ∈ S ⟹ open_segment a x ⊆ rel_interior S" using rel_interior_closure_convex_segment a ‹convex S› closure_subset subsetCE by blast have starT: "⋀x. x ∈ T ⟹ open_segment b x ⊆ rel_interior T" using rel_interior_closure_convex_segment b ‹convex T› closure_subset subsetCE by blast let ?aS = "(+) (-a) ` S" and ?bT = "(+) (-b) ` T" have 0: "0 ∈ affine hull ?aS" "0 ∈ affine hull ?bT" by (metis a b subsetD hull_inc image_eqI left_minus rel_interior_subset)+ have subs: "subspace (span ?aS)" "subspace (span ?bT)" by (rule subspace_span)+ moreover have "dim (span ((+) (- a) ` S)) = dim (span ((+) (- b) ` T))" by (metis 0 aff_dim_translation_eq aff_dim_zero affeq dim_span nat_int) ultimately obtain f g where "linear f" "linear g" and fim: "f ` span ?aS = span ?bT" and gim: "g ` span ?bT = span ?aS" and fno: "⋀x. x ∈ span ?aS ⟹ norm(f x) = norm x" and gno: "⋀x. x ∈ span ?bT ⟹ norm(g x) = norm x" and gf: "⋀x. x ∈ span ?aS ⟹ g(f x) = x" and fg: "⋀x. x ∈ span ?bT ⟹ f(g x) = x" by (rule isometries_subspaces) blast have [simp]: "continuous_on A f" for A using ‹linear f› linear_conv_bounded_linear linear_continuous_on by blast have [simp]: "continuous_on B g" for B using ‹linear g› linear_conv_bounded_linear linear_continuous_on by blast have eqspanS: "affine hull ?aS = span ?aS" by (metis a affine_hull_span_0 subsetD hull_inc image_eqI left_minus rel_interior_subset) have eqspanT: "affine hull ?bT = span ?bT" by (metis b affine_hull_span_0 subsetD hull_inc image_eqI left_minus rel_interior_subset) have "S homeomorphic cball a 1 ∩ affine hull S" by (rule starlike_compact_projective2 [OF ‹compact S› a starS]) also have "... homeomorphic (+) (-a) ` (cball a 1 ∩ affine hull S)" by (metis homeomorphic_translation) also have "... = cball 0 1 ∩ (+) (-a) ` (affine hull S)" by (auto simp: dist_norm) also have "... = cball 0 1 ∩ span ?aS" using eqspanS affine_hull_translation by blast also have "... homeomorphic cball 0 1 ∩ span ?bT" proof (rule homeomorphicI) show fim1: "f ` (cball 0 1 ∩ span ?aS) = cball 0 1 ∩ span ?bT" proof show "f ` (cball 0 1 ∩ span ?aS) ⊆ cball 0 1 ∩ span ?bT" using fim fno by auto show "cball 0 1 ∩ span ?bT ⊆ f ` (cball 0 1 ∩ span ?aS)" by clarify (metis IntI fg gim gno image_eqI mem_cball_0) qed show "g ` (cball 0 1 ∩ span ?bT) = cball 0 1 ∩ span ?aS" proof show "g ` (cball 0 1 ∩ span ?bT) ⊆ cball 0 1 ∩ span ?aS" using gim gno by auto show "cball 0 1 ∩ span ?aS ⊆ g ` (cball 0 1 ∩ span ?bT)" by clarify (metis IntI fim1 gf image_eqI) qed qed (auto simp: fg gf) also have "... = cball 0 1 ∩ (+) (-b) ` (affine hull T)" using eqspanT affine_hull_translation by blast also have "... = (+) (-b) ` (cball b 1 ∩ affine hull T)" by (auto simp: dist_norm) also have "... homeomorphic (cball b 1 ∩ affine hull T)" by (metis homeomorphic_translation homeomorphic_sym) also have "... homeomorphic T" by (metis starlike_compact_projective2 [OF ‹compact T› b starT] homeomorphic_sym) finally have 1: "S homeomorphic T" . have "S - rel_interior S homeomorphic sphere a 1 ∩ affine hull S" by (rule starlike_compact_projective1 [OF ‹compact S› a starS]) also have "... homeomorphic (+) (-a) ` (sphere a 1 ∩ affine hull S)" by (metis homeomorphic_translation) also have "... = sphere 0 1 ∩ (+) (-a) ` (affine hull S)" by (auto simp: dist_norm) also have "... = sphere 0 1 ∩ span ?aS" using eqspanS affine_hull_translation by blast also have "... homeomorphic sphere 0 1 ∩ span ?bT" proof (rule homeomorphicI) show fim1: "f ` (sphere 0 1 ∩ span ?aS) = sphere 0 1 ∩ span ?bT" proof show "f ` (sphere 0 1 ∩ span ?aS) ⊆ sphere 0 1 ∩ span ?bT" using fim fno by auto show "sphere 0 1 ∩ span ?bT ⊆ f ` (sphere 0 1 ∩ span ?aS)" by clarify (metis IntI fg gim gno image_eqI mem_sphere_0) qed show "g ` (sphere 0 1 ∩ span ?bT) = sphere 0 1 ∩ span ?aS" proof show "g ` (sphere 0 1 ∩ span ?bT) ⊆ sphere 0 1 ∩ span ?aS" using gim gno by auto show "sphere 0 1 ∩ span ?aS ⊆ g ` (sphere 0 1 ∩ span ?bT)" by clarify (metis IntI fim1 gf image_eqI) qed qed (auto simp: fg gf) also have "... = sphere 0 1 ∩ (+) (-b) ` (affine hull T)" using eqspanT affine_hull_translation by blast also have "... = (+) (-b) ` (sphere b 1 ∩ affine hull T)" by (auto simp: dist_norm) also have "... homeomorphic (sphere b 1 ∩ affine hull T)" by (metis homeomorphic_translation homeomorphic_sym) also have "... homeomorphic T - rel_interior T" by (metis starlike_compact_projective1 [OF ‹compact T› b starT] homeomorphic_sym) finally have 2: "S - rel_interior S homeomorphic T - rel_interior T" . show ?thesis using 1 2 by blast qed lemma homeomorphic_convex_compact_sets: fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" assumes "convex S" "compact S" "convex T" "compact T" and affeq: "aff_dim S = aff_dim T" shows "S homeomorphic T" using homeomorphic_convex_lemma [OF assms] assms by (auto simp: rel_frontier_def) lemma homeomorphic_rel_frontiers_convex_bounded_sets: fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" assumes "convex S" "bounded S" "convex T" "bounded T" and affeq: "aff_dim S = aff_dim T" shows "rel_frontier S homeomorphic rel_frontier T" using assms homeomorphic_convex_lemma [of "closure S" "closure T"] by (simp add: rel_frontier_def convex_rel_interior_closure) subsection‹Homeomorphisms between punctured spheres and affine sets› text‹Including the famous stereoscopic projection of the 3-D sphere to the complex plane› text‹The special case with centre 0 and radius 1› lemma homeomorphic_punctured_affine_sphere_affine_01: assumes "b ∈ sphere 0 1" "affine T" "0 ∈ T" "b ∈ T" "affine p" and affT: "aff_dim T = aff_dim p + 1" shows "(sphere 0 1 ∩ T) - {b} homeomorphic p" proof - have [simp]: "norm b = 1" "b∙b = 1" using assms by (auto simp: norm_eq_1) have [simp]: "T ∩ {v. b∙v = 0} ≠ {}" using ‹0 ∈ T› by auto have [simp]: "¬ T ⊆ {v. b∙v = 0}" using ‹norm b = 1› ‹b ∈ T› by auto define f where "f ≡ λx. 2 *⇩_{R}b + (2 / (1 - b∙x)) *⇩_{R}(x - b)" define g where "g ≡ λy. b + (4 / (norm y ^ 2 + 4)) *⇩_{R}(y - 2 *⇩_{R}b)" have fg[simp]: "⋀x. ⟦x ∈ T; b∙x = 0⟧ ⟹ f (g x) = x" unfolding f_def g_def by (simp add: algebra_simps field_split_simps add_nonneg_eq_0_iff) have no: "(norm (f x))⇧^{2}= 4 * (1 + b ∙ x) / (1 - b ∙ x)" if "norm x = 1" and "b ∙ x ≠ 1" for x using that sum_sqs_eq [of 1 "b ∙ x"] apply (simp flip: dot_square_norm add: norm_eq_1 nonzero_eq_divide_eq) apply (simp add: f_def vector_add_divide_simps inner_simps) apply (auto simp add: field_split_simps inner_commute) done have [simp]: "⋀u::real. 8 + u * (u * 8) = u * 16 ⟷ u=1" by algebra have gf[simp]: "⋀x. ⟦norm x = 1; b ∙ x ≠ 1⟧ ⟹ g (f x) = x" unfolding g_def no by (auto simp: f_def field_split_simps) have g1: "norm (g x) = 1" if "x ∈ T" and "b ∙ x = 0" for x using that apply (simp only: g_def) apply (rule power2_eq_imp_eq) apply (simp_all add: dot_square_norm [symmetric] divide_simps vector_add_divide_simps) apply (simp add: algebra_simps inner_commute) done have ne1: "b ∙ g x ≠ 1" if "x ∈ T" and "b ∙ x = 0" for x using that unfolding g_def apply (simp_all add: dot_square_norm [symmetric] divide_simps vector_add_divide_simps add_nonneg_eq_0_iff) apply (auto simp: algebra_simps) done have "subspace T" by (simp add: assms subspace_affine) have gT: "⋀x. ⟦x ∈ T; b ∙ x = 0⟧ ⟹ g x ∈ T" unfolding g_def by (blast intro: ‹subspace T› ‹b ∈ T› subspace_add subspace_mul subspace_diff) have "f ` {x. norm x = 1 ∧ b∙x ≠ 1} ⊆ {x. b∙x = 0}" unfolding f_def using ‹norm b = 1› norm_eq_1 by (force simp: field_simps inner_add_right inner_diff_right) moreover have "f ` T ⊆ T" unfolding f_def using assms ‹subspace T› by (auto simp add: inner_add_right inner_diff_right mem_affine_3_minus subspace_mul) moreover have "{x. b∙x = 0} ∩ T ⊆ f ` ({x. norm x = 1 ∧ b∙x ≠ 1} ∩ T)" by clarify (metis (mono_tags) IntI ne1 fg gT g1 imageI mem_Collect_eq) ultimately have imf: "f ` ({x. norm x = 1 ∧ b∙x ≠ 1} ∩ T) = {x. b∙x = 0} ∩ T" by blast have no4: "⋀y. b∙y = 0 ⟹ norm ((y∙y + 4) *⇩_{R}b + 4 *⇩_{R}(y - 2 *⇩_{R}b)) = y∙y + 4" apply (rule power2_eq_imp_eq) apply (simp_all flip: dot_square_norm) apply (auto simp: power2_eq_square algebra_simps inner_commute) done have [simp]: "⋀x. ⟦norm x = 1; b ∙ x ≠ 1⟧ ⟹ b ∙ f x = 0" by (simp add: f_def algebra_simps field_split_simps) have [simp]: "⋀x. ⟦x ∈ T; norm x = 1; b ∙ x ≠ 1⟧ ⟹ f x ∈ T" unfolding f_def by (blast intro: ‹subspace T› ‹b ∈ T› subspace_add subspace_mul subspace_diff) have "g ` {x. b∙x = 0} ⊆ {x. norm x = 1 ∧ b∙x ≠ 1}" unfolding g_def apply (clarsimp simp: no4 vector_add_divide_simps divide_simps add_nonneg_eq_0_iff dot_square_norm [symmetric]) apply (auto simp: algebra_simps) done moreover have "g ` T ⊆ T" unfolding g_def by (blast intro: ‹subspace T› ‹b ∈ T› subspace_add subspace_mul subspace_diff) moreover have "{x. norm x = 1 ∧ b∙x ≠ 1} ∩ T ⊆ g ` ({x. b∙x = 0} ∩ T)" by clarify (metis (mono_tags, lifting) IntI gf image_iff imf mem_Collect_eq) ultimately have img: "g ` ({x. b∙x = 0} ∩ T) = {x. norm x = 1 ∧ b∙x ≠ 1} ∩ T" by blast have aff: "affine ({x. b∙x = 0} ∩ T)" by (blast intro: affine_hyperplane assms) have contf: "continuous_on ({x. norm x = 1 ∧ b∙x ≠ 1} ∩ T) f" unfolding f_def by (rule continuous_intros | force)+ have contg: "continuous_on ({x. b∙x = 0} ∩ T) g" unfolding g_def by (rule continuous_intros | force simp: add_nonneg_eq_0_iff)+ have "(sphere 0 1 ∩ T) - {b} = {x. norm x = 1 ∧ (b∙x ≠ 1)} ∩ T" using ‹norm b = 1› by (auto simp: norm_eq_1) (metis vector_eq ‹b∙b = 1›) also have "... homeomorphic {x. b∙x = 0} ∩ T" by (rule homeomorphicI [OF imf img contf contg]) auto also have "... homeomorphic p" proof (rule homeomorphic_affine_sets [OF aff ‹affine p›]) show "aff_dim ({x. b ∙ x = 0} ∩ T) = aff_dim p" by (simp add: Int_commute aff_dim_affine_Int_hyperplane [OF ‹affine T›] affT) qed finally show ?thesis . qed theorem homeomorphic_punctured_affine_sphere_affine: fixes a :: "'a :: euclidean_space" assumes "0 < r" "b ∈ sphere a r" "affine T" "a ∈ T" "b ∈ T" "affine p" and aff: "aff_dim T = aff_dim p + 1" shows "(sphere a r ∩ T) - {b} homeomorphic p" proof - have "a ≠ b" using assms by auto then have inj: "inj (λx::'a. x /⇩_{R}norm (a - b))" by (simp add: inj_on_def) have "((sphere a r ∩ T) - {b}) homeomorphic (+) (-a) ` ((sphere a r ∩ T) - {b})" by (rule homeomorphic_translation) also have "... homeomorphic (*⇩_{R}) (inverse r) ` (+) (- a) ` (sphere a r ∩ T - {b})" by (metis ‹0 < r› homeomorphic_scaling inverse_inverse_eq inverse_zero less_irrefl) also have "... = sphere 0 1 ∩ ((*⇩_{R}) (inverse r) ` (+) (- a) ` T) - {(b - a) /⇩_{R}r}" using assms by (auto simp: dist_norm norm_minus_commute divide_simps) also have "... homeomorphic p" using assms affine_translation [symmetric, of "- a"] aff_dim_translation_eq [of "- a"] by (intro homeomorphic_punctured_affine_sphere_affine_01) (auto simp: dist_norm norm_minus_commute affine_scaling inj) finally show ?thesis . qed corollary homeomorphic_punctured_sphere_affine: fixes a :: "'a :: euclidean_space" assumes "0 < r" and b: "b ∈ sphere a r" and "affine T" and affS: "aff_dim T + 1 = DIM('a)" shows "(sphere a r - {b}) homeomorphic T" using homeomorphic_punctured_affine_sphere_affine [of r b a UNIV T] assms by auto corollary homeomorphic_punctured_sphere_hyperplane: fixes a :: "'a :: euclidean_space" assumes "0 < r" and b: "b ∈ sphere a r" and "c ≠ 0" shows "(sphere a r - {b}) homeomorphic {x::'a. c ∙ x = d}" using assms by (intro homeomorphic_punctured_sphere_affine) (auto simp: affine_hyperplane of_nat_diff) proposition homeomorphic_punctured_sphere_affine_gen: fixes a :: "'a :: euclidean_space" assumes "convex S" "bounded S" and a: "a ∈ rel_frontier S" and "affine T" and affS: "aff_dim S = aff_dim T + 1" shows "rel_frontier S - {a} homeomorphic T" proof - obtain U :: "'a set" where "affine U" "convex U" and affdS: "aff_dim U = aff_dim S" using choose_affine_subset [OF affine_UNIV aff_dim_geq] by (meson aff_dim_affine_hull affine_affine_hull affine_imp_convex) have "S ≠ {}" using assms by auto then obtain z where "z ∈ U" by (metis aff_dim_negative_iff equals0I affdS) then have bne: "ball z 1 ∩ U ≠ {}" by force then have [simp]: "aff_dim(ball z 1 ∩ U) = aff_dim U" using aff_dim_convex_Int_open [OF ‹convex U› open_ball] by (fastforce simp add: Int_commute) have "rel_frontier S homeomorphic rel_frontier (ball z 1 ∩ U)" by (rule homeomorphic_rel_frontiers_convex_bounded_sets) (auto simp: ‹affine U› affine_imp_convex convex_Int affdS assms) also have "... = sphere z 1 ∩ U" using convex_affine_rel_frontier_Int [of "ball z 1" U] by (simp add: ‹affine U› bne) finally have "rel_frontier S homeomorphic sphere z 1 ∩ U" . then obtain h k where him: "h ` rel_frontier S = sphere z 1 ∩ U" and kim: "k ` (sphere z 1 ∩ U) = rel_frontier S" and hcon: "continuous_on (rel_frontier S) h" and kcon: "continuous_on (sphere z 1 ∩ U) k" and kh: "⋀x. x ∈ rel_frontier S ⟹ k(h(x)) = x" and hk: "⋀y. y ∈ sphere z 1 ∩ U ⟹ h(k(y)) = y" unfolding homeomorphic_def homeomorphism_def by auto have "rel_frontier S - {a} homeomorphic (sphere z 1 ∩ U) - {h a}" proof (rule homeomorphicI) show h: "h ` (rel_frontier S - {a}) = sphere z 1 ∩ U - {h a}" using him a kh by auto metis show "k ` (sphere z 1 ∩ U - {h a}) = rel_frontier S - {a}" by (force simp: h [symmetric] image_comp o_def kh) qed (auto intro: continuous_on_subset hcon kcon simp: kh hk) also have "... homeomorphic T" by (rule homeomorphic_punctured_affine_sphere_affine) (use a him in ‹auto simp: affS affdS ‹affine T› ‹affine U› ‹z ∈ U››) finally show ?thesis . qed text‹ When dealing with AR, ANR and ANR later, it's useful to know that every set is homeomorphic to a closed subset of a convex set, and if the set is locally compact we can take the convex set to be the universe.› proposition homeomorphic_closedin_convex: fixes S :: "'m::euclidean_space set" assumes "aff_dim S < DIM('n)" obtains U and T :: "'n::euclidean_space set" where "convex U" "U ≠ {}" "closedin (top_of_set U) T" "S homeomorphic T" proof (cases "S = {}") case True then show ?thesis by (rule_tac U=UNIV and T="{}" in that) auto next case False then obtain a where "a ∈ S" by auto obtain i::'n where i: "i ∈ Basis" "i ≠ 0" using SOME_Basis Basis_zero by force have "0 ∈ affine hull ((+) (- a) ` S)" by (simp add: ‹a ∈ S› hull_inc) then have "dim ((+) (- a) ` S) = aff_dim ((+) (- a) ` S)" by (simp add: aff_dim_zero) also have "... < DIM('n)" by (simp add: aff_dim_translation_eq_subtract assms cong: image_cong_simp) finally have dd: "dim ((+) (- a) ` S) < DIM('n)" by linarith have span: "span {x. i ∙ x = 0} = {x. i ∙ x = 0}" using span_eq_iff [symmetric, of "{x. i ∙ x = 0}"] subspace_hyperplane [of i] by simp have "dim ((+) (- a) ` S) ≤ dim {x. i ∙ x = 0}" using dd by (simp add: dim_hyperplane [OF ‹i ≠ 0›]) then obtain T where "subspace T" and Tsub: "T ⊆ {x. i ∙ x = 0}" and dimT: "dim T = dim ((+) (- a) ` S)" by (rule choose_subspace_of_subspace) (simp add: span) have "subspace (span ((+) (- a) ` S))" using subspace_span by blast then obtain h k where "linear h" "linear k" and heq: "h ` span ((+) (- a) ` S) = T" and keq:"k ` T = span ((+) (- a) ` S)" and hinv [simp]: "⋀x. x ∈ span ((+) (- a) ` S) ⟹ k(h x) = x" and kinv [simp]: "⋀x. x ∈ T ⟹ h(k x) = x" by (auto simp: dimT intro: isometries_subspaces [OF _ ‹subspace T›] dimT) have hcont: "continuous_on A h" and kcont: "continuous_on B k" for A B using ‹linear h› ‹linear k› linear_continuous_on linear_conv_bounded_linear by blast+ have ihhhh[simp]: "⋀x. x ∈ S ⟹ i ∙ h (x - a) = 0" using Tsub [THEN subsetD] heq span_superset by fastforce have "sphere 0 1 - {i} homeomorphic {x. i ∙ x = 0}" proof (rule homeomorphic_punctured_sphere_affine) show "affine {x. i ∙ x = 0}" by (auto simp: affine_hyperplane) show "aff_dim {x. i ∙ x = 0} + 1 = int DIM('n)" using i by clarsimp (metis DIM_positive Suc_pred add.commute of_nat_Suc) qed (use i in auto) then obtain f g where fg: "homeomorphism (sphere 0 1 - {i}) {x. i ∙ x = 0} f g" by (force simp: homeomorphic_def) show ?thesis proof have "h ` (+) (- a) ` S ⊆ T" using heq span_superset span_linear_image by blast then have "g ` h ` (+) (- a) ` S ⊆ g ` {x. i ∙ x = 0}" using Tsub by (simp add: image_mono) also have "... ⊆ sphere 0 1 - {i}" by (simp add: fg [unfolded homeomorphism_def]) finally have gh_sub_sph: "(g ∘ h) ` (+) (- a) ` S ⊆ sphere 0 1 - {i}" by (metis image_comp) then have gh_sub_cb: "(g ∘ h) ` (+) (- a) ` S ⊆ cball 0 1" by (metis Diff_subset order_trans sphere_cball) have [simp]: "⋀u. u ∈ S ⟹ norm (g (h (u - a))) = 1" using gh_sub_sph [THEN subsetD] by (auto simp: o_def) show "convex (ball 0 1 ∪ (g ∘ h) ` (+) (- a) ` S)" by (meson ball_subset_cball convex_intermediate_ball gh_sub_cb sup.bounded_iff sup.cobounded1) show "closedin (top_of_set (ball 0 1 ∪ (g ∘ h) ` (+) (- a) ` S)) ((g ∘ h) ` (+) (- a) ` S)" unfolding closedin_closed by (rule_tac x="sphere 0 1" in exI) auto have ghcont: "continuous_on ((λx. x - a) ` S) (λx. g (h x))" by (rule continuous_on_compose2 [OF homeomorphism_cont2 [OF fg] hcont], force) have kfcont: "continuous_on ((λx. g (h (x - a))) ` S) (λx. k (f x))" proof (rule continuous_on_compose2 [OF kcont]) show "continuous_on ((λx. g (h (x - a))) ` S) f" using homeomorphism_cont1 [OF fg] gh_sub_sph by (fastforce intro: continuous_on_subset) qed auto have "S homeomorphic (+) (- a) ` S" by (fact homeomorphic_translation) also have "… homeomorphic (g ∘ h) ` (+) (- a) ` S" apply (simp add: homeomorphic_def homeomorphism_def cong: image_cong_simp) apply (rule_tac x="g ∘ h" in exI) apply (rule_tac x="k ∘ f" in exI) apply (auto simp: ghcont kfcont span_base homeomorphism_apply2 [OF fg] image_comp cong: image_cong_simp) done finally show "S homeomorphic (g ∘ h) ` (+) (- a) ` S" . qed auto qed subsection‹Locally compact sets in an open set› text‹ Locally compact sets are closed in an open set and are homeomorphic to an absolutely closed set if we have one more dimension to play with.› lemma locally_compact_open_Int_closure: fixes S :: "'a :: metric_space set" assumes "locally compact S" obtains T where "open T" "S = T ∩ closure S" proof - have "∀x∈S. ∃T v u. u = S ∩ T ∧ x ∈ u ∧ u ⊆ v ∧ v ⊆ S ∧ open T ∧ compact v" by (metis assms locally_compact openin_open) then obtain t v where tv: "⋀x. x ∈ S ⟹ v x ⊆ S ∧ open (t x) ∧ compact (v x) ∧ (∃u. x ∈ u ∧ u ⊆ v x ∧ u = S ∩ t x)" by metis then have o: "open (⋃(t ` S))" by blast have "S = ⋃ (v ` S)" using tv by blast also have "... = ⋃(t ` S) ∩ closure S" proof show "⋃(v ` S) ⊆ ⋃(t ` S) ∩ closure S" by clarify (meson IntD2 IntI UN_I closure_subset subsetD tv) have "t x ∩ closure S ⊆ v x" if "x ∈ S" for x proof - have "t x ∩ closure S ⊆ closure (t x ∩ S)" by (simp add: open_Int_closure_subset that tv) also have "... ⊆ v x" by (metis Int_commute closure_minimal compact_imp_closed that tv) finally show ?thesis . qed then show "⋃(t ` S) ∩ closure S ⊆ ⋃(v ` S)" by blast qed finally have e: "S = ⋃(t ` S) ∩ closure S" . show ?thesis by (rule that [OF o e]) qed lemma locally_compact_closedin_open: fixes S :: "'a :: metric_space set" assumes "locally compact S" obtains T where "open T" "closedin (top_of_set T) S" by (metis locally_compact_open_Int_closure [OF assms] closed_closure closedin_closed_Int) lemma locally_compact_homeomorphism_projection_closed: assumes "locally compact S" obtains T and f :: "'a ⇒ 'a :: euclidean_space × 'b :: euclidean_space" where "closed T" "homeomorphism S T f fst" proof (cases "closed S") case True show ?thesis proof show "homeomorphism S (S × {0}) (λx. (x, 0)) fst" by (auto simp: homeomorphism_def continuous_intros) qed (use True closed_Times in auto) next case False obtain U where "open U" and US: "U ∩ closure S = S" by (metis locally_compact_open_Int_closure [OF assms]) with False have Ucomp: "-U ≠ {}" using closure_eq by auto have [simp]: "closure (- U) = -U" by (simp add: ‹open U› closed_Compl) define f :: "'a ⇒ 'a × 'b" where "f ≡ λx. (x, One /⇩_{R}setdist {x} (- U))" have "continuous_on U (λx. (x, One /⇩_{R}setdist {x} (- U)))" proof (intro continuous_intros continuous_on_setdist) show "∀x∈U. setdist {x} (- U) ≠ 0" by (simp add: Ucomp setdist_eq_0_sing_1) qed then have homU: "homeomorphism U (f`U) f fst" by (auto simp: f_def homeomorphism_def image_iff continuous_intros) have cloS: "closedin (top_of_set U) S" by (metis US closed_closure closedin_closed_Int) have cont: "isCont ((λx. setdist {x} (- U)) o fst) z" for z :: "'a × 'b" by (rule continuous_at_compose continuous_intros continuous_at_setdist)+ have setdist1D: "setdist {a} (- U) *⇩_{R}b = One ⟹ setdist {a} (- U) ≠ 0" for a::'a and b::'b by force have *: "r *⇩_{R}b = One ⟹ b = (1 / r) *⇩_{R}One" for r and b::'b by (metis One_non_0 nonzero_divide_eq_eq real_vector.scale_eq_0_iff real_vector.scale_scale scaleR_one) have "⋀a b::'b. setdist {a} (- U) *⇩_{R}b = One ⟹ (a,b) ∈ (λx. (x, (1 / setdist {x} (- U)) *⇩_{R}One)) ` U" by (metis (mono_tags, lifting) "*" ComplI image_eqI setdist1D setdist_sing_in_set) then have "f ` U = (λz. (setdist {fst z} (- U) *⇩_{R}snd z)) -` {One}" by (auto simp: f_def setdist_eq_0_sing_1 field_simps Ucomp) then have clfU: "closed (f ` U)" by (force intro: continuous_intros cont [unfolded o_def] continuous_closed_vimage) have "closed (f ` S)" by (metis closedin_closed_trans [OF _ clfU] homeomorphism_imp_closed_map [OF homU cloS]) then show ?thesis by (metis US homU homeomorphism_of_subsets inf_sup_ord(1) that) qed lemma locally_compact_closed_Int_open: fixes S :: "'a :: euclidean_space set" shows "locally compact S ⟷ (∃U V. closed U ∧ open V ∧ S = U ∩ V)" (is "?lhs = ?rhs") proof show "?lhs ⟹ ?rhs" by (metis closed_closure inf_commute locally_compact_open_Int_closure) show "?rhs ⟹ ?lhs" by (meson closed_imp_locally_compact locally_compact_Int open_imp_locally_compact) qed lemma lowerdim_embeddings: assumes "DIM('a) < DIM('b)" obtains f :: "'a::euclidean_space*real ⇒ 'b::euclidean_space" and g :: "'b ⇒ 'a*real" and j :: 'b where "linear f" "linear g" "⋀z. g (f z) = z" "j ∈ Basis" "⋀x. f(x,0) ∙ j = 0" proof - let ?B = "Basis :: ('a*real) set" have b01: "(0,1) ∈ ?B" by (simp add: Basis_prod_def) have "DIM('a * real) ≤ DIM('b)" by (simp add: Suc_leI assms) then obtain basf :: "'a*real ⇒ 'b" where sbf: "basf ` ?B ⊆ Basis" and injbf: "inj_on basf Basis" by (metis finite_Basis card_le_inj) define basg:: "'b ⇒ 'a * real" where "basg ≡ λi. if i ∈ basf ` Basis then inv_into Basis basf i else (0,1)" have bgf[simp]: "basg (basf i) = i" if "i ∈ Basis" for i using inv_into_f_f injbf that by (force simp: basg_def) have sbg: "basg ` Basis ⊆ ?B" by (force simp: basg_def injbf b01) define f :: "'a*real ⇒ 'b" where "f ≡ λu. ∑j∈Basis. (u ∙ basg j) *⇩_{R}j" define g :: "'b ⇒ 'a*real" where "g ≡ λz. (∑i∈Basis. (z ∙ basf i) *⇩_{R}i)" show ?thesis proof show "linear f" unfolding f_def by (intro linear_compose_sum linearI ballI) (auto simp: algebra_simps) show "linear g" unfolding g_def by (intro linear_compose_sum linearI ballI) (auto simp: algebra_simps) have *: "(∑a ∈ Basis. a ∙ basf b * (x ∙ basg a)) = x ∙ b" if "b ∈ Basis" for x b using sbf that by auto show gf: "g (f x) = x" for x proof (rule euclidean_eqI) show "⋀b. b ∈ Basis ⟹ g (f x) ∙ b = x ∙ b" using f_def g_def sbf by auto qed show "basf(0,1) ∈ Basis" using b01 sbf by auto then show "f(x,0) ∙ basf(0,1) = 0" for x unfolding f_def inner_sum_left using b01 inner_not_same_Basis by (fastforce intro: comm_monoid_add_class.sum.neutral) qed qed proposition locally_compact_homeomorphic_closed: fixes S :: "'a::euclidean_space set" assumes "locally compact S" and dimlt: "DIM('a) < DIM('b)" obtains T :: "'b::euclidean_space set" where "closed T" "S homeomorphic T" proof - obtain U:: "('a*real)set" and h where "closed U" and homU: "homeomorphism S U h fst" using locally_compact_homeomorphism_projection_closed assms by metis obtain f :: "'a*real ⇒ 'b" and g :: "'b ⇒ 'a*real" where "linear f" "linear g" and gf [simp]: "⋀z. g (f z) = z" using lowerdim_embeddings [OF dimlt] by metis then have "inj f" by (metis injI) have gfU: "g ` f ` U = U" by (simp add: image_comp o_def) have "S homeomorphic U" using homU homeomorphic_def by blast also have "... homeomorphic f ` U" proof (rule homeomorphicI [OF refl gfU]) show "continuous_on U f" by (meson ‹inj f› ‹linear f› homeomorphism_cont2 linear_homeomorphism_image) show "continuous_on (f ` U) g" using ‹linear g› linear_continuous_on linear_conv_bounded_linear by blast qed (auto simp: o_def) finally show ?thesis using ‹closed U› ‹inj f› ‹linear f› closed_injective_linear_image that by blast qed lemma homeomorphic_convex_compact_lemma: fixes S :: "'a::euclidean_space set" assumes "convex S" and "compact S" and "cball 0 1 ⊆ S" shows "S homeomorphic (cball (0::'a) 1)" proof (rule starlike_compact_projective_special[OF assms(2-3)]) fix x u assume "x ∈ S" and "0 ≤ u" and "u < (1::real)" have "open (ball (u *⇩_{R}x) (1 - u))" by (rule open_ball) moreover have "u *⇩_{R}x ∈ ball (u *⇩_{R}x) (1 - u)" unfolding centre_in_ball using ‹u < 1› by simp moreover have "ball (u *⇩_{R}x) (1 - u) ⊆ S" proof fix y assume "y ∈ ball (u *⇩_{R}x) (1 - u)" then have "dist (u *⇩_{R}x) y < 1 - u" unfolding mem_ball . with ‹u < 1› have "inverse (1 - u) *⇩_{R}(y - u *⇩_{R}x) ∈ cball 0 1" by (simp add: dist_norm inverse_eq_divide norm_minus_commute) with assms(3) have "inverse (1 - u) *⇩_{R}(y - u *⇩_{R}x) ∈ S" .. with assms(1) have "(1 - u) *⇩_{R}((y - u *⇩_{R}x) /⇩_{R}(1 - u)) + u *⇩_{R}x ∈ S" using ‹x ∈ S› ‹0 ≤ u› ‹u < 1› [THEN less_imp_le] by (rule convexD_alt) then show "y ∈ S" using ‹u < 1› by simp qed ultimately have "u *⇩_{R}x ∈ interior S" .. then show "u *⇩_{R}x ∈ S - frontier S" using frontier_def and interior_subset by auto qed proposition homeomorphic_convex_compact_cball: fixes e :: real and S :: "'a::euclidean_space set" assumes S: "convex S" "compact S" "interior S ≠ {}" and "e > 0" shows "S homeomorphic (cball (b::'a) e)" proof (rule homeomorphic_trans[OF _ homeomorphic_balls(2)]) obtain a where "a ∈ interior S" using assms by auto then show "S homeomorphic cball (0::'a) 1" by (metis (no_types) aff_dim_cball S compact_cball convex_cball homeomorphic_convex_lemma interior_rel_interior_gen zero_less_one) qed (use ‹e>0› in auto) corollary homeomorphic_convex_compact: fixes S :: "'a::euclidean_space set" and T :: "'a set" assumes "convex S" "compact S" "interior S ≠ {}" and "convex T" "compact T" "interior T ≠ {}" shows "S homeomorphic T" using assms by (meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym) lemma homeomorphic_closed_intervals: fixes a :: "'a::euclidean_space" and b and c :: "'a::euclidean_space" and d assumes "box a b ≠ {}" and "box c d ≠ {}" shows "(cbox a b) homeomorphic (cbox c d)" by (simp add: assms homeomorphic_convex_compact) lemma homeomorphic_closed_intervals_real: fixes a::real and b and c::real and d assumes "a<b" and "c<d" shows "{a..b} homeomorphic {c..d}" using assms by (auto intro: homeomorphic_convex_compact) subsection‹Covering spaces and lifting results for them› definition✐‹tag important› covering_space :: "'a::topological_space set ⇒ ('a ⇒ 'b) ⇒ 'b::topological_space set ⇒ bool" where "covering_space c p S ≡ continuous_on c p ∧ p ` c = S ∧ (∀x ∈ S. ∃T. x ∈ T ∧ openin (top_of_set S) T ∧ (∃v. ⋃v = c ∩ p -` T ∧ (∀u ∈ v. openin (top_of_set c) u) ∧ pairwise disjnt v ∧ (∀u ∈ v. ∃q. homeomorphism u T p q)))" lemma covering_spaceI [intro?]: assumes "continuous_on c p" "p ` c = S" "⋀x. x ∈ S ⟹ ∃T. x ∈ T ∧ openin (top_of_set S) T ∧ (∃v. ⋃v = c ∩ p -` T ∧ (∀u ∈ v. openin (top_of_set c) u) ∧ pairwise disjnt v ∧ (∀u ∈ v. ∃q. homeomorphism u T p q))" shows "covering_space c p S" using assms unfolding covering_space_def by auto lemma covering_space_imp_continuous: "covering_space c p S ⟹ continuous_on c p" by (simp add: covering_space_def) lemma covering_space_imp_surjective: "covering_space c p S ⟹ p ` c = S" by (simp add: covering_space_def) lemma homeomorphism_imp_covering_space: "homeomorphism S T f g ⟹ covering_space S f T" apply (clarsimp simp add: homeomorphism_def covering_space_def) apply (rule_tac x=T in exI, simp) apply (rule_tac x="{S}" in exI, auto) done lemma covering_space_local_homeomorphism: assumes "covering_space c p S" "x ∈ c" obtains T u q where "x ∈ T" "openin (top_of_set c) T" "p x ∈ u" "openin (top_of_set S) u" "homeomorphism T u p q" using assms by (clarsimp simp add: covering_space_def) (metis IntI UnionE vimage_eq) lemma covering_space_local_homeomorphism_alt: assumes p: "covering_space c p S" and "y ∈ S" obtains x T U q where "p x = y" "x ∈ T" "openin (top_of_set c) T" "y ∈ U" "openin (top_of_set S) U" "homeomorphism T U p q" proof - obtain x where "p x = y" "x ∈ c" using assms covering_space_imp_surjective by blast show ?thesis using that ‹p x = y› by (auto intro: covering_space_local_homeomorphism [OF p ‹x ∈ c›]) qed proposition covering_space_open_map: fixes S :: "'a :: metric_space set" and T :: "'b :: metric_space set" assumes p: "covering_space c p S" and T: "openin (top_of_set c) T" shows "openin (top_of_set S) (p ` T)" proof - have pce: "p ` c = S" and covs: "⋀x. x ∈ S ⟹ ∃X VS. x ∈ X ∧ openin (top_of_set S) X ∧ ⋃VS = c ∩ p -` X ∧ (∀u ∈ VS. openin (top_of_set c) u) ∧ pairwise disjnt VS ∧ (∀u ∈ VS. ∃q. homeomorphism u X p q)" using p by (auto simp: covering_space_def) have "T ⊆ c" by (metis openin_euclidean_subtopology_iff T) have "∃X. openin (top_of_set S) X ∧ y ∈ X ∧ X ⊆ p ` T" if "y ∈ p ` T" for y proof - have "y ∈ S" using ‹T ⊆ c› pce that by blast obtain U VS where "y ∈ U" and U: "openin (top_of_set S) U" and VS: "⋃VS = c ∩ p -` U" and openVS: "∀V ∈ VS. openin (top_of_set c) V" and homVS: "⋀V. V ∈ VS ⟹ ∃q. homeomorphism V U p q" using covs [OF ‹y ∈ S›] by auto obtain x where "x ∈ c" "p x ∈ U" "x ∈ T" "p x = y" using T [unfolded openin_euclidean_subtopology_iff] ‹y ∈ U› ‹y ∈ p ` T› by blast with VS obtain V where "x ∈ V" "V ∈ VS" by auto then obtain q where q: "homeomorphism V U p q" using homVS by blast then have ptV: "p ` (T ∩ V) = U ∩ q -` (T ∩ V)" using VS ‹V ∈ VS› by (auto simp: homeomorphism_def) have ocv: "openin (top_of_set c) V" by (simp add: ‹V ∈ VS› openVS) have "openin (top_of_set (q ` U)) (T ∩ V)" using q unfolding homeomorphism_def by (metis T inf.absorb_iff2 ocv openin_imp_subset openin_subtopology_Int subtopology_subtopology) then have "openin (top_of_set U) (U ∩ q -` (T ∩ V))" using continuous_on_open homeomorphism_def q by blast then have os: "openin (top_of_set S) (U ∩ q -` (T ∩ V))" using openin_trans [of U] by (simp add: Collect_conj_eq U) show ?thesis proof (intro exI conjI) show "openin (top_of_set S) (p ` (T ∩ V))" by (simp only: ptV os) qed (use ‹p x = y› ‹x ∈ V› ‹x ∈ T› in auto) qed with openin_subopen show ?thesis by blast qed lemma covering_space_lift_unique_gen: fixes f :: "'a::topological_space ⇒ 'b::topological_space" fixes g1 :: "'a ⇒ 'c::real_normed_vector" assumes cov: "covering_space c p S" and eq: "g1 a = g2 a" and f: "continuous_on T f" "f ∈ T → S" and g1: "continuous_on T g1" "g1 ∈ T → c" and fg1: "⋀x. x ∈ T ⟹ f x = p(g1 x)" and g2: "continuous_on T g2" "g2 ∈ T → c" and fg2: "⋀x. x ∈ T ⟹ f x = p(g2 x)" and u_compt: "U ∈ components T" and "a ∈ U" "x ∈ U" shows "g1 x = g2 x" proof - have "U ⊆ T" by (rule in_components_subset [OF u_compt]) define G12 where "G12 ≡ {x ∈ U. g1 x - g2 x = 0}" have "connected U" by (rule in_components_connected [OF u_compt]) have contu: "continuous_on U g1" "continuous_on U g2" using ‹U ⊆ T› continuous_on_subset g1 g2 by blast+ have o12: "openin (top_of_set U) G12" unfolding G12_def proof (subst openin_subopen, clarify) fix z assume z: "z ∈ U" "g1 z - g2 z = 0" obtain v w q where "g1 z ∈ v" and ocv: "openin (top_of_set c) v" and "p (g1 z) ∈ w" and osw: "openin (top_of_set S) w" and hom: "homeomorphism v w p q" proof (rule covering_space_local_homeomorphism [OF cov]) show "g1 z ∈ c" using ‹U ⊆ T› ‹z ∈ U› g1(2) by blast qed auto have "g2 z ∈ v" using ‹g1 z ∈ v› z by auto have gg: "U ∩ g -` v = U ∩ g -` (v ∩ g ` U)" for g by auto have "openin (top_of_set (g1 ` U)) (v ∩ g1 ` U)" using ocv ‹U ⊆ T› g1 by (fastforce simp add: openin_open) then have 1: "openin (top_of_set U) (U ∩ g1 -` v)" unfolding gg by (blast intro: contu continuous_on_open [THEN iffD1, rule_format]) have "openin (top_of_set (g2 ` U)) (v ∩ g2 ` U)" using ocv ‹U ⊆ T› g2 by (fastforce simp add: openin_open) then have 2: "openin (top_of_set U) (U ∩ g2 -` v)" unfolding gg by (blast intro: contu continuous_on_open [THEN iffD1,