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.
[1] 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.
[1] A countable collection of subsets need not be locally finite, as shown by the collection of all subsets of
for a natural number n. Every locally finite collection of sets is point finite, meaning that every point of the space belongs to only finitely many sets in the collection.
Point finiteness is a strictly weaker notion, as illustrated by the collection of intervals
If a collection of sets is locally finite, the collection of the closures of these sets is also locally finite.
[3] 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
However, the union of a locally finite collection of closed sets is closed.
is a point outside the union of this locally finite collection of closed sets, we merely choose a neighbourhood
that intersects this collection at only finitely many of these sets.
Define a bijective map from the collection of sets that
thus giving an index to each of these sets.
that does not intersect the union of this collection of closed sets.
be a locally finite family of subsets of a compact space
, choose an open neighbourhood
that intersects a finite number of the subsets in
intersects only a finite number of subsets in
intersects only a finite number of subsets in
intersects only a finite number of subsets in the collection
Every locally finite collection of sets in a Lindelöf space, in particular in a second-countable space, is countable.
[5] This is proved by a similar argument as in the result above for compact spaces.
A collection of subsets of a topological space is called σ-locally finite[6][7] or countably locally finite[8] if it is a countable union of locally finite collections.
The σ-locally finite notion is a key ingredient in the Nagata–Smirnov metrization theorem, which states that a topological space is metrizable if and only if it is regular, Hausdorff, and has a σ-locally finite base.
[9][10] In a Lindelöf space, in particular in a second-countable space, every σ-locally finite collection of sets is countable.