(* This file is a part of IsarMathLib - a library of formalized mathematics for Isabelle/Isar. Copyright (C) 2005, 2006 Slawomir Kolodynski This program is free software; Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The name of the author may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. *) section ‹Topology 1b› theory Topology_ZF_1b imports Topology_ZF_1 begin text‹One of the facts demonstrated in every class on General Topology is that in a $T_2$ (Hausdorff) topological space compact sets are closed. Formalizing the proof of this fact gave me an interesting insight into the role of the Axiom of Choice (AC) in many informal proofs. A typical informal proof of this fact goes like this: we want to show that the complement of $K$ is open. To do this, choose an arbitrary point $y\in K^c$. Since $X$ is $T_2$, for every point $x\in K$ we can find an open set $U_x$ such that $y\notin \overline{U_x}$. Obviously $\{U_x\}_{x\in K}$ covers $K$, so select a finite subcollection that covers $K$, and so on. I had never realized that such reasoning requires the Axiom of Choice. Namely, suppose we have a lemma that states "In $T_2$ spaces, if $x\neq y$, then there is an open set $U$ such that $x\in U$ and $y\notin \overline{U}$" (like our lemma ‹T2_cl_open_sep› below). This only states that the set of such open sets $U$ is not empty. To get the collection $\{U_x \}_{x\in K}$ in this proof we have to select one such set among many for every $x\in K$ and this is where we use the Axiom of Choice. Probably in 99/100 cases when an informal calculus proof states something like $\forall \varepsilon \exists \delta_\varepsilon \cdots$ the proof uses AC. Most of the time the use of AC in such proofs can be avoided. This is also the case for the fact that in a $T_2$ space compact sets are closed. › subsection‹Compact sets are closed - no need for AC› text‹In this section we show that in a $T_2$ topological space compact sets are closed.› text‹First we prove a lemma that in a $T_2$ space two points can be separated by the closure of an open set.› lemma (in topology0) T2_cl_open_sep: assumes "T {is T⇩_{2}}" and "x ∈ ⋃T" "y ∈ ⋃T" "x≠y" shows "∃U∈T. (x∈U ∧ y ∉ cl(U))" proof - from assms have "∃U∈T. ∃V∈T. x∈U ∧ y∈V ∧ U∩V=0" using isT2_def by simp then obtain U V where "U∈T" "V∈T" "x∈U" "y∈V" "U∩V=0" by auto then have "U∈T ∧ x∈U ∧ y∈ V ∧ cl(U) ∩ V = 0" using disj_open_cl_disj by auto thus "∃U∈T. (x∈U ∧ y ∉ cl(U))" by auto qed text‹AC-free proof that in a Hausdorff space compact sets are closed. To understand the notation recall that in Isabelle/ZF ‹Pow(A)› is the powerset (the set of subsets) of $A$ and ‹FinPow(A)› denotes the set of finite subsets of $A$ in IsarMathLib.› theorem (in topology0) in_t2_compact_is_cl: assumes A1: "T {is T⇩_{2}}" and A2: "K {is compact in} T" shows "K {is closed in} T" proof - let ?X = "⋃T" have "∀y ∈ ?X - K. ∃U∈T. y∈U ∧ U ⊆ ?X - K" proof - { fix y assume "y ∈ ?X" "y∉K" have "∃U∈T. y∈U ∧ U ⊆ ?X - K" proof - let ?B = "⋃x∈K. {V∈T. x∈V ∧ y ∉ cl(V)}" have I: "?B ∈ Pow(T)" "FinPow(?B) ⊆ Pow(?B)" using FinPow_def by auto from ‹K {is compact in} T› ‹y ∈ ?X› ‹y∉K› have "∀x∈K. x ∈ ?X ∧ y ∈ ?X ∧ x≠y" using IsCompact_def by auto with ‹T {is T⇩_{2}}› have "∀x∈K. {V∈T. x∈V ∧ y ∉ cl(V)} ≠ 0" using T2_cl_open_sep by auto hence "K ⊆ ⋃?B" by blast with ‹K {is compact in} T› I have "∃N ∈ FinPow(?B). K ⊆ ⋃N" using IsCompact_def by auto then obtain N where "N ∈ FinPow(?B)" "K ⊆ ⋃N" by auto with I have "N ⊆ ?B" by auto hence "∀V∈N. V∈?B" by auto let ?M = "{cl(V). V∈N}" let ?C = "{D ∈ Pow(?X). D {is closed in} T}" from ‹N ∈ FinPow(?B)› have "∀V∈?B. cl(V) ∈ ?C" "N ∈ FinPow(?B)" using cl_is_closed IsClosed_def by auto then have "?M ∈ FinPow(?C)" by (rule fin_image_fin) then have "?X - ⋃?M ∈ T" using fin_union_cl_is_cl IsClosed_def by simp moreover from ‹y ∈ ?X› ‹y∉K› ‹∀V∈N. V∈?B› have "y ∈ ?X - ⋃?M" by simp moreover have "?X - ⋃?M ⊆ ?X - K" proof - from ‹∀V∈N. V∈?B› have "⋃N ⊆ ⋃?M" using cl_contains_set by auto with ‹K ⊆ ⋃N› show "?X - ⋃?M ⊆ ?X - K" by auto qed ultimately have "∃U. U∈T ∧ y ∈ U ∧ U ⊆ ?X - K" by auto thus "∃U∈T. y∈U ∧ U ⊆ ?X - K" by auto qed } thus "∀y ∈ ?X - K. ∃U∈T. y∈U ∧ U ⊆ ?X - K" by auto qed with A2 show "K {is closed in} T" using open_neigh_open IsCompact_def IsClosed_def by auto qed end