Properties
Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact. This can be extended to F-sigma subspaces as well.
- A regular space is paracompact if every open cover admits a locally finite refinement. (Here, the refinement is not required to be open.) In particular, every regular Lindelof 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.
Read more about this topic: Paracompact Space
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