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 formula_1 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 formula_2 of the form formula_3 for an integer formula_4. A countable collection of subsets need not be locally finite, as shown by the collection of all subsets of formula_2 of the form formula_6 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 formula_2 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 formula_2 and the empty set).
Compact spaces.
Every locally finite collection of subsets of a compact space must be finite. Indeed, let formula_9 be an locally finite family of subsets of a compact space formula_1 . For each point formula_11, choose an open neighbourhood formula_12 that intersects a finite number of the subsets in formula_13. Clearly the family of sets: formula_14 is an open cover of formula_1, and therefor has a finite subcover: formula_16. Since each formula_17 intersects only a finite number of subsets in formula_13, the union of all such formula_17 intersects only a finite number of subsets in formula_13. Since this union is the whole space formula_1, it follows that formula_22 intersects only a finite number of subsets in the collection formula_13. And since formula_13 is composed of subsets of formula_1 every member of formula_13 must intersect formula_1, thus formula_13 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 formula_29 is a point outside the union of this locally finite collection of closed sets, we merely choose a neighbourhood formula_30 of formula_29 that intersects this collection at only finitely many of these sets. Define a bijective map from the collection of sets that formula_30 intersects to formula_33 thus giving an index to each of these sets. Then for each set, choose an open set formula_34 containing formula_29 that doesn't intersect it. The intersection of all such formula_34 for formula_37 intersected with formula_30, is a neighbourhood of formula_29 that does not intersect the union of this collection of closed sets.
Countably locally finite collections.
A collection in a space formula_1 is countably locally finite (or σ-locally finite) if it is the union of a countable family of locally finite collections of subsets of formula_1. 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.