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Concept
Locally finite collection
Résumé
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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