compact pavings are closed subsets of a compact space
Recall that a paving is compact if every subcollection satisfying the finite intersection property has nonempty intersection. In particular, a topological space is compact (http://planetmath.org/Compact) if and only if its collection 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 the following result says that the collection of all intersections of finite unions of sets in is also compact.
Suppose that is a compact paved space. Let be the smallest collection of subsets of such that and which is closed under arbitrary intersections and finite unions. Then, is a compact paving.
is closed under arbitrary unions and finite intersections, and hence is a topology on . The collection of closed sets defined with respect to this topology is which, by Theorem 1, is a compact paving. So, the following is obtained.
A paving is compact if and only if there exists a topology on with respect to which are closed sets and is compact.
|Title||compact pavings are closed subsets of a compact space|
|Date of creation||2013-03-22 18:45:01|
|Last modified on||2013-03-22 18:45:01|
|Last modified by||gel (22282)|