Locally finite collection
Encyclopedia
In the mathematical
field of topology
, local finiteness is a property of collections of subset
s 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 locally finite has different meanings in other mathematical fields.
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 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).
can be locally finite. Indeed, let {Ga} 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 Ux that intersects the collection {Ga} at only finitely many values of a. Clearly:
and hence has a finite subcover, Ua1 ∪ ...... ∪ Uan. Since each Uai intersects {Ga} for only finitely many values of a, the union of all such Uai intersects the collection {Ga} for only finitely many values of a. It follows that X (the whole space!) intersects the collection {Ga} at only finitely many values of a contradicting the infinite cardinality of the collection {Ga}.
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 X is also point-finite. A topological space in which every open cover admits a point-finite open refinement is called metacompact
.
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.
, which states that a topological space is metrizable if and only if it is regular
Hausdorff
and has a countably locally finite basis.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, local finiteness is a property of collections of subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
. 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
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
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 R of the form (n, n + 2) with integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
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 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).
Compact spaces
No infinite collection of a compact spaceCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
can be locally finite. Indeed, let {Ga} 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 Ux that intersects the collection {Ga} at only finitely many values of a. Clearly:
- Ux for each x in X (the union over all x) is a open covering in X
and hence has a finite subcover, Ua1 ∪ ...... ∪ Uan. Since each Uai intersects {Ga} for only finitely many values of a, the union of all such Uai intersects the collection {Ga} for only finitely many values of a. It follows that X (the whole space!) intersects the collection {Ga} at only finitely many values of a contradicting the infinite cardinality of the collection {Ga}.
A topological space in which every open cover admits a locally finite open refinement is called paracompact
Paracompact space
In mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by ....
. Every locally finite collection of subsets of a topological space X is also point-finite. A topological space in which every open cover admits a point-finite open refinement is called metacompact
Metacompact space
In mathematics, in the field of general topology, a topological space is said to be metacompact if every open cover has a point finite open refinement...
.
Second countable spaces
No uncountable coverCover (topology)
In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset. Formally, ifC = \lbrace U_\alpha: \alpha \in A\rbrace...
of a Lindelöf space
Lindelöf space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
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
Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
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 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 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 Ui than doesn't intersect it. The intersection of all such Ui for 1 ≤ i ≤ k intersected with V, is a neighbourhood of x that does not intersect the union of this collection of closed sets.Countably locally 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 theoremNagata–Smirnov metrization theorem
The Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space X is metrizable if and only if it is regular and Hausdorff and has a countably locally finite basis.Unlike Urysohn's metrization theorem, which...
, which states that a topological space is metrizable if and only if it is regular
Regular space
In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...
Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
and has a countably locally finite basis.