Topology Atlas Document # paca-03

Proof of a lemma on paracompactness plus applications

Henno Brandsma

a note in Topology Explained
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We will prove a previously stated lemma on paracompactness:

For a regular space X the following are equivalent:

  1. X is paracompact, i.e. every open cover of X has a locally finite open refinement.

  2. Every open cover of X has an open countably locally finite refinement.

  3. Every open cover of X has a locally finite refinement.

  4. Every open cover of X has a locally finite closed refinement.

Date: April 1, 2003

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