Metacompact space

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In mathematics, in the field of 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.

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.