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

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*)

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 T2}"  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 T2}" 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 T2}› 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