Editing Paracompact space (section)

== Properties ==

Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact. This can be extended to [[F-sigma set|F-sigma]] subspaces as well.{{sfn | Dugundji | 1966 | pp=165, Theorem 2.2}}

* ([[Michael's theorem on paracompact spaces|Michael's theorem]]) A [[regular space]] is paracompact if every open cover admits a locally finite refinement, not necessarily open. In particular, every regular [[Lindelöf space]] is paracompact.
* ('''Smirnov metrization theorem''') A topological space is metrizable if and only if it is paracompact, Hausdorff, and locally metrizable.
* [[Michael selection theorem]] states that lower semicontinuous multifunctions from ''X'' into nonempty closed convex subsets of Banach spaces admit continuous selection iff ''X'' is paracompact.

Although a product of paracompact spaces need not be paracompact, the following are true:

* The product of a paracompact space and a [[compact space]] is paracompact.
* The product of a [[metacompact space]] and a compact space is metacompact.

Both these results can be proved by the [[tube lemma]] which is used in the proof that a product of ''finitely many'' compact spaces is compact.