|
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. == 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 (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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Metacompact space」の詳細全文を読む スポンサード リンク
|