A collection of subsets of a topological space is said to be locally finite if each point in the space has a neighbourhood that intersects only finitely many of the sets in the collection.
In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension.
Note that the term locally finite has different meanings in other mathematical fields.
A finite collection of subsets of a topological space is locally finite. Infinite collections can also be locally finite: for example, the collection of all subsets of of the form for an integer . A countable collection of subsets need not be locally finite, as shown by the collection of all subsets of of the form for a natural number n.
If a collection of sets is locally finite, the collection of all closures of these sets is also locally finite. The reason for this is that if an open set containing a point intersects the closure of a set, it necessarily intersects the set itself, hence a neighborhood can intersect at most the same number of closures (it may intersect fewer, since two distinct, indeed disjoint, sets can have the same closure). The converse, however, can fail if the closures of the sets are not distinct. For example, in the finite complement topology on the collection of all open sets is not locally finite, but the collection of all closures of these sets is locally finite (since the only closures are and the empty set).
Every locally finite collection of subsets of a compact space must be finite. Indeed, let be a locally finite family of subsets of a compact space . For each point , choose an open neighbourhood that intersects a finite number of the subsets in . Clearly the family of sets: is an open cover of , and therefore has a finite subcover: . Since each intersects only a finite number of subsets in , the union of all such intersects only a finite number of subsets in .
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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.
In mathematics, a collection or family of subsets of a topological space is said to be point-finite if every point of lies in only finitely many members of A metacompact space is a topological space in which every open cover admits a point-finite open refinement. Every locally finite collection of subsets of a topological space is also point-finite. A topological space in which every open cover admits a locally finite open refinement is called a paracompact space. Every paracompact space is therefore metacompact.
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology. All spaces in this glossary are assumed to be topological spaces unless stated otherwise. Absolutely closed See H-closed Accessible See . Accumulation point See limit point.
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