Point Finite Collection
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High Quality Content by WIKIPEDIA articles! 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 topological space, X, is said to be locally finite if every collection of subsets of it is locally finite. Since every locally finite collection is point finite, every collection of subsets of X mu...
High Quality Content by WIKIPEDIA articles! 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 topological space, X, is said to be locally finite if every collection of subsets of it is locally finite. Since every locally finite collection is point finite, every collection of subsets of X must be point-finite. In particular, this means that the power set of this space must be finite (if it were infinite, the collection of all subsets of the space wouldn't be locally finite since some point would belong to infinitely many subsets of this space). This means that the space is finite. Hence a topological space is locally finite if and only if it is finite.
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