(*

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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*)

header{*\isaheader{Topology\_ZF\_1b.thy}*}

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 @{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;

qed

text{*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;

qed

end