proof of compact pavings are closed subsets of a compact space

Let $(K,\mathcal{K})$ be a compact paved space (http://planetmath.org/paved space). We use the ultrafilter lemma (http://planetmath.org/EveryFilterIsContainedInAnUltrafilter) to show that there is a compact paving $\mathcal{K}^{\prime}$ containing $\mathcal{K}$ that is closed under arbitrary intersections and finite unions.

We first show that the paving $\mathcal{K}_{1}$ consisting of all finite unions of elements of $\mathcal{K}$ is compact. Let $\mathcal{F}\subseteq\mathcal{K}_{1}$ satisfy the finite intersection property. It then follows that the collection of finite intersections of $\mathcal{F}$ is a filter (http://planetmath.org/Filter). The ultrafilter lemma says that $\mathcal{F}$ is contained in an ultrafilter $\mathcal{U}$ .

By definition, the ultrafilter satisfies the finite intersection property. So, the compactness of $\mathcal{K}$ implies that $\mathcal{F}^{\prime}\equiv\mathcal{U}\cap\mathcal{K}$ has nonempty intersection. Also, every element $S$ of $\mathcal{F}$ is a union of finitely many elements of $\mathcal{K}$ , one of which must be in $\mathcal{U}$ (see alternative characterization of ultrafilter (http://planetmath.org/AlternativeCharacterizationOfUltrafilter)). In particular, $S$ contains the intersection of $\mathcal{F}^{\prime}$ and,

$\bigcap\mathcal{F}\supseteq\bigcap\mathcal{F}^{\prime}\not=\emptyset.$

Consequently, $\mathcal{K}_{1}$ is compact.

Finally, we let $\mathcal{K}^{\prime}$ be the set of arbitrary intersections of $\mathcal{K}_{1}$ . This is closed under all arbitrary intersections and finite unions. Furthermore, if $\mathcal{F}\subseteq\mathcal{K}^{\prime}$ satisfies the finite intersection property then so does

$\mathcal{F}^{\prime}\equiv\left\{A\in\mathcal{K}_{1}\colon B\subseteq A\text{ % for some }B\in\mathcal{F}\right\}.$

The compactness of $\mathcal{K}_{1}$ gives

$\bigcap\mathcal{F}=\bigcap\mathcal{F}^{\prime}\not=\emptyset$

as required.

Title	proof of compact pavings are closed subsets of a compact space
Canonical name	ProofOfCompactPavingsAreClosedSubsetsOfACompactSpace
Date of creation	2013-03-22 18:45:07
Last modified on	2013-03-22 18:45:07
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	4
Author	gel (22282)
Entry type	Proof
Classification	msc 28A05