compact pavings are closed subsets of a compact space

Recall that a paving 𝒦 is compactPlanetmathPlanetmath if every subcollection satisfying the finite intersection property has nonempty intersectionMathworldPlanetmath. In particular, a topological spaceMathworldPlanetmath is compact ( if and only if its collectionMathworldPlanetmath of closed subsets forms a compact paving. Compact paved spaces can therefore be constructed by taking closed subsets of a compact topological space. In fact, all compact pavings arise in this way, as we now show.

Given any compact paving 𝒦 the following result says that the collection 𝒦 of all intersections of finite unions of sets in 𝒦 is also compact.

Theorem 1.

Suppose that (K,K) is a compact paved space. Let K be the smallest collection of subsets of X such that KK and which is closed under arbitrary intersections and finite unions. Then, K is a compact paving.

In particular,


is closed under arbitrary unions and finite intersections, and hence is a topology on K. The collection of closed sets defined with respect to this topology is 𝒦{,K} which, by Theorem 1, is a compact paving. So, the following is obtained.


A paving (K,K) is compact if and only if there exists a topology on K with respect to which K are closed sets and K is compact.

Title compact pavings are closed subsets of a compact space
Canonical name CompactPavingsAreClosedSubsetsOfACompactSpace
Date of creation 2013-03-22 18:45:01
Last modified on 2013-03-22 18:45:01
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Theorem
Classification msc 28A05