Metacompact space

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In mathematics, in the field of general topology, a topological space is said to be metacompact if every open cover has a point finite open refinement. That is, given any open cover of the topological space, there is a refinement which is again an open cover with the property that every point is contained only in finitely many sets of the refining cover.

A space is countably metacompact if every countable open cover has a point finite open refinement.

The following can be said about metacompactness in relation to other properties of topological spaces:

Every paracompact space is metacompact.
Every metacompact space is orthocompact.
Every metacompact normal space is a shrinking space
The product of a compact space and a metacompact space is metacompact. This follows from the tube lemma.