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.

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.

Note that the term locally finite has different meanings in other mathematical fields.

Examples and properties

1. A finite collection in a topological space is locally finite, but the converse need not hold. For instance, the collection of all subsets of R of the form (n, n + 2) is locally finite but not finite. However, R itself is not locally finite, as the collection of all subsets of the form (−n, n) for n a positive integer is not a locally finite collection. In particular, this latter example shows that a countable collection need not be locally finite. An uncountable collection of subsets of a topological space may or may not be locally finite.

2. A nice example is to show that if a collection of sets is locally finite, the collection of all closures of these sets need not be locally finite. The only way that this can happen is if the closures of these sets aren't distinct. An example of this is the finite complement topology on R. The collection of all open sets isn't locally finite, but the collection of all closures of these sets is locally finite (since each open set is dense, the closure of each open set is R so that the collection of closures of all open sets consists of exactly one element).

Compact spaces

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:

U_x for each x in X (the union over all x) is a open covering in X

and hence has a finite subcover, U_a₁ ∪ ...... ∪ U_{a_n}. Since each U_{a_i} intersects {G_a} for only finitely many values of a, the union of all such U_{a_i} intersects the collection {G_a} for only finitely many values of a. It follows that X (the whole space!) intersects the collection {G_a} at only finitely many values of a contradicting the infinite cardinality of the collection {G_a}.

Second countable spaces

No uncountable cover of a Lindelöf space space can be locally finite, by essentially the same argument as in the case of compact spaces. In particular, no uncountable cover of a second-countable space is locally finite.

Closed sets

It is clear from the definition of a topology that a finite union of closed sets is closed. One can readily give an example of an infinite union of closed sets that isn't closed. However, if we consider a locally finite collection of closed sets, the union is closed. To see this we note that if x is a point outside the union of this locally finite collection of closed sets, we merely choose a neighbourhood V of x that intersects this collection at only finitely many of these sets. Define a bijective map from the collection of sets that V intersects to {1, ..., k} thus giving an index to each of these sets. Then for each set, choose an open set U_i than doesn't intersect it. The intersection of all such U_i for 1 ≤ i ≤ k intersected with V, is a neighbourhood of x which doesn't intersect the union of this collection of closed sets.

Countably local finite collections

A collection in a space is countably locally finite (or σ-locally finite) if it is the union of a countable family of locally finite collections of subsets of X. Countable local finiteness is a key hypothesis in the Nagata–Smirnov metrization theorem which states that a topological space is metrizable if and only if it is regular and has a countably locally finite basis.

References

Topology 2ed, James R. Munkres, Prentice Hall