Abstract:
A space is compact if every open cover of it has a finite subcover,
paracompact if every open cover has an open refinement which is locally finite
and metacompact if every open cover has a refinement which is point finite.
These properties can be "selectivized" in various ways. For example: One
selectivization of compactness is: For each sequence of open covers of the space
there is a sequence of finite sets, the n-th a subset of the n-th cover, such that the
union of these finite sets is a cover of the space. This is the classical Menger
property, Sfin(O,O).