# Theory Topology_ZF_1b

theory Topology_ZF_1b
imports Topology_ZF_1
(*     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 APARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT,INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOTLIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; ORBUSINESS 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 THEUSE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.*)header{*\isaheader{Topology\_ZF\_1b.thy}*}theory Topology_ZF_1b imports Topology_ZF_1begintext{*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 @{text "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. *}section{*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 ∈ \<Union>T"  "y ∈ \<Union>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;qedtext{*AC-free proof that in a Hausdorff space compact sets   are closed. To understand the notation recall that in Isabelle/ZF  @{text "Pow(A)"} is the powerset (the set of subsets) of $A$   and @{text "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 = "\<Union>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 = "\<Union>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 ⊆ \<Union>?B" by blast;	with K {is compact in} T I have 	  "∃N ∈ FinPow(?B). K ⊆ \<Union>N" 	  using IsCompact_def by auto;	then obtain N where "N ∈ FinPow(?B)"  "K ⊆ \<Union>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 - \<Union>?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 - \<Union>?M" by simp;	moreover have "?X - \<Union>?M ⊆ ?X - K"	proof -	  from ∀V∈N. V∈?B have "\<Union>N ⊆ \<Union>?M" using cl_contains_set by auto;	  with K ⊆ \<Union>N show "?X - \<Union>?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;qedend