Locally finite 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.

A collection of subsets of a topological space X 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.

Note that the term 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 R of the form (n, n + 2) with integer n. A countable collection of subsets need not be locally finite, as shown by the collection of all subsets of R of the form (−n, n) with integer 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 R 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 R and the empty set).

No infinite collection of a compact space can be locally finite. Indeed, let {G_a} be an infinite family of subsets of a space and suppose this collection is locally finite. For each point x of this space, choose a neighbourhood U_x that intersects the collection {G_a} at only finitely many values of a. Clearly:

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