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 ${\displaystyle X}$ 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 locally finite has different meanings in other mathematical fields.

Examples and properties

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 ${\displaystyle \mathbb {R} }$ of the form ${\displaystyle (n,n+2)}$ for an integer ${\displaystyle n}$. A countable collection of subsets need not be locally finite, as shown by the collection of all subsets of ${\displaystyle \mathbb {R} }$ of the form ${\displaystyle (-n,n)}$ 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 ${\displaystyle \mathbb {R} }$ 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 ${\displaystyle \mathbb {R} }$ and the empty set).

Compact spaces

Every locally finite collection of subsets of a compact space must be finite. Indeed, let ${\displaystyle G=\{G_{a}|a\in A\}}$ be an locally finite family of subsets of a compact space ${\displaystyle X}$ . For each point ${\displaystyle x\in X}$, choose an open neighbourhood ${\displaystyle U_{x}}$ that intersects a finite number of the subsets in ${\displaystyle G}$. Clearly the family of sets: ${\displaystyle \{U_{x}|x\in X\}}$ is an open cover of ${\displaystyle X}$, and therefor has a finite subcover: ${\displaystyle \{U_{k_{n}}|n\in 1\dots n\}}$. Since each ${\displaystyle U_{k_{i}}}$ intersects only a finite number of subsets in ${\displaystyle G}$, the union of all such ${\displaystyle U_{k_{i}}}$ intersects only a finite number of subsets in ${\displaystyle G}$. Since this union is the whole space ${\displaystyle X}$, it follows that intersects only a finite number of subsets in the collection ${\displaystyle G}$. And since ${\displaystyle G}$ is composed of subsets of ${\displaystyle X}$ every member of ${\displaystyle G}$ must intersect ${\displaystyle X}$, thus ${\displaystyle G}$ is finite.

A topological space in which every open cover admits a locally finite open refinement is called paracompact. Every locally finite collection of subsets of a topological space is also point-finite. A topological space in which every open cover admits a point-finite open refinement is called metacompact.

Second countable spaces

No uncountable cover of a Lindelöf 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

A finite union of closed sets is always closed. One can readily give an example of an infinite union of closed sets that is not closed. However, if we consider a locally finite collection of closed sets, the union is closed. To see this we note that if ${\displaystyle x}$ is a point outside the union of this locally finite collection of closed sets, we merely choose a neighbourhood ${\displaystyle V}$ of ${\displaystyle x}$ that intersects this collection at only finitely many of these sets. Define a bijective map from the collection of sets that ${\displaystyle V}$ intersects to ${\displaystyle {1,\dots ,k}}$ thus giving an index to each of these sets. Then for each set, choose an open set ${\displaystyle U_{i}}$ containing ${\displaystyle x}$ that doesn't intersect it. The intersection of all such ${\displaystyle U_{i}}$ for ${\displaystyle 1\leq i\leq k}$ intersected with ${\displaystyle V}$, is a neighbourhood of ${\displaystyle x}$ that does not intersect the union of this collection of closed sets.

Countably locally finite collections

A collection in a space ${\displaystyle X}$ is countably locally finite (or σ-locally finite) if it is the union of a countable family of locally finite collections of subsets of ${\displaystyle X}$. 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

• James R. Munkres (2000), Topology (2nd ed.), Prentice Hall, ISBN 0-13-181629-2