Metacompact space

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 (but a countably 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 (see Watson). == Covering dimension ==

A topological space X is said to be of covering dimension n if every open cover of X has a point-finite open refinement such that no point of X is included in more than n + 1 sets in the refinement and if n is the minimum value for which this is true. If no such minimal n exists, the space is said to be of infinite covering dimension. ==See also==