HOME

TheInfoList



OR:

In the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, local finiteness is a property of collections of
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
. It is fundamental in the study of paracompactness and
topological dimension In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. Informal discussion For ordinary Euclidean ...
. A collection of subsets of a topological space X is said to be locally finite if each point in the space has a
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
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 \mathbb of the form (n, n+2) for an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
n. A
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
collection of subsets need not be locally finite, as shown by the collection of all subsets of \mathbb of the form (-n, n) for a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
''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 In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
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 \mathbb 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 \mathbb and the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
).


Compact spaces

Every locally finite collection of subsets of a
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
must be finite. Indeed, let G=\ be a locally finite family of subsets of a compact space X . For each point x\in X, choose an
open neighbourhood In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a ...
U_ that intersects a finite number of the subsets in G. Clearly the family of sets: \ is an
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alp ...
of X, and therefore has a finite subcover: \. Since each U_ intersects only a finite number of subsets in G, the union of all such U_ intersects only a finite number of subsets in G. Since this union is the whole space X, it follows that intersects only a finite number of subsets in the collection G. And since G is composed of subsets of X every member of G must intersect X, thus G is finite. A topological space in which every
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alp ...
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 Cover or covers may refer to: Packaging * Another name for a lid * Cover (philately), generic term for envelope or package * Album cover, the front of the packaging * Book cover or magazine cover ** Book design ** Back cover copy, part of copy ...
of a
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'' su ...
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 set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
s 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 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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
map from the collection of sets that V intersects to thus giving an index to each of these sets. Then for each set, choose an open set U_i containing x that doesn't intersect it. The intersection of all such U_i for 1\leq i\leq 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 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 X. Countably local finiteness is a key hypothesis in the
Nagata–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, Hausdorff and has a countably locally finite ( ...
, which states that a topological space is metrizable if and only if it is regular and has a countably locally finite basis.


References

*{{Citation, title=Topology, edition=2nd, author=James R. Munkres, publisher=Prentice Hall, year=2000, isbn=0-13-181629-2 General topology Properties of topological spaces