Metacompact space
In the mathematical 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 that 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.
Properties
The following can be said about metacompactness in relation to other properties of topological spaces:- Every paracompact space is metacompact. This implies that every compact space is metacompact, and every metric space is metacompact. The converse does not hold: a counter-example is the Dieudonné plank.
- 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.
- An easy example of a non-metacompact space is the Moore plane.
- In order for a Tychonoff space X to be compact it is necessary and sufficient that X be metacompact and pseudocompact.