In
topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
and related branches of
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
topological spaceTopological 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...
is called
locally compact if, roughly speaking, each small portion of the space looks like a small portion of a
compact spaceIn mathematics, more specifically general topology and metric topology, a compact space is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space...
.
Formal definition
Let
X be a
topological spaceTopological 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...
. The following are common definitions for
X is locally compact, and are
equivalent if X is a Hausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T
2 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...
(or preregular). They are
not equivalent in general:
- 1. every point of X has a compact neighbourhood.
- 2. every point of X has a closed
In topology and related branches of mathematics, a closed set is a set whose complement is open.- Equivalent definitions of a closed set :In a topological space, a set is closed if and only if it coincides with its closure...
compact neighbourhood.
- 2‘. every point has a relatively compact neighbourhood.
- 2‘‘. every point has a local base of relatively compact neighbourhoods.
- 3. every point of X has a local base of compact neighbourhoods.
Logical relations among the conditions:
- Conditions (2), (2‘), (2‘‘) are equivalent.
- Neither of conditions (2), (3) implies the other.
- Each condition implies (1).
- Compactness implies conditions (1) and (2), but not (3).
Condition (1) is probably the most commonly used definition, since it is the least restrictive and the others are equivalent to it when
X is
HausdorffIn topology and related branches of mathematics, a Hausdorff space, separated space or T
2 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...
. This equivalence is a consequence of the facts that compact subsets of Hausdorff spaces are closed, and closed subsets of compact spaces are compact.
Authors such as Munkres and Kelley use the first definition. Willard uses the third. In Steen and Seebach, a space which satisfies (1) is said to be
locally compact, while a space satisfying (2) is said to be
strongly locally compact.
In almost all applications, locally compact spaces are also Hausdorff, and this article is thus primarily concerned with locally compact Hausdorff (LCH) spaces.
Compact Hausdorff spaces
Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article
compact spaceIn mathematics, more specifically general topology and metric topology, a compact space is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space...
.
Here we mention only:
- the unit interval
In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
[0,1];
- any closed topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
;
- the Cantor set
In mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883 , is a set of points lying on a single line segment that has a number of remarkable and deep properties. Through consideration of it, Cantor and others helped lay the foundations of modern general topology...
;
- the Hilbert cube
In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology...
.
Locally compact Hausdorff spaces that are not compact
- The Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
s Rn (and in particular the real lineIn mathematics, the real line is the line whose points correspond to the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space...
R) are locally compact as a consequence of the Heine-Borel theorem.
- Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
s share the local properties of Euclidean spaces and are therefore also all locally compact. This even includes nonparacompact manifolds such as the long lineIn topology, the long line is a topological space analogous to the real line, but much longer. Because it behaves locally just like the real line, but has different large-scale properties, it serves as one of the basic counterexamples of topology.- Definition :The closed long ray L is defined as...
.
- All discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
s are locally compact and Hausdorff (they are just the zero0 is both a number and the numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, zero is used as a placeholder in place value systems...
-dimensional manifolds). These are compact only if they are finite.
- All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset of , the subspace...
. This provides several examples of locally compact subsets of Euclidean spaces, such as the unit disc (either the open or closed version).
- The space Qp of p-adic numbers
In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. For each prime number p, the p-adic number system extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
is locally compact, because it is homeomorphic to the Cantor setIn mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883 , is a set of points lying on a single line segment that has a number of remarkable and deep properties. Through consideration of it, Cantor and others helped lay the foundations of modern general topology...
minus one point. Thus locally compact spaces are as useful in p-adic analysisIn mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers.The theory of complex-valued numerical functions on the p-adic numbers is just part of the theory of locally compact groups...
as in classical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
.
Hausdorff spaces that are not locally compact
As mentioned in the following section, no Hausdorff space can possibly be locally compact if it is not also a
Tychonoff spaceIn topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces.These conditions are examples of separation axioms....
; there are some examples of Hausdorff spaces that are not Tychonoff spaces in that article.
But there are also examples of Tychonoff spaces that fail to be locally compact, such as:
- the space Q of rational number
In mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...
s (endowed with the topology from R), since its compact subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
s all have empty interiorIn mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S". A point that is in the interior of S is an interior point of S....
and therefore are not neighborhoods;
- the subspace {(0,0)} union
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets gives a set .- Definition :A simple example:...
{(x,y) : x > 0} of R2, since the origin does not have a compact neighborhood;
- the lower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties...
or upper limit topology on the set R of real numbers (useful in the study of one-sided limitIn calculus, a one-sided limit is either of the two limits of a function f of a real variable x as x approaches a specified point either from below or from above...
s);
- any T0, hence Hausdorff, topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
that is infinite-dimensionIn mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al, such as an infinite-dimensional Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
.
The first two examples show that a subset of a locally compact space need not be locally compact, which contrasts with the open and closed subsets in the previous section.
The last example contrasts with the Euclidean spaces in the previous section; to be more specific, a Hausdorff topological vector space is locally compact if and only if it is finite-dimensional (in which case it is a Euclidean space).
This example also contrasts with the Hilbert cube as an example of a compact space; there is no contradiction because the cube cannot be a neighbourhood of any point in Hilbert space.
Non-Hausdorff examples
- The one-point compactification of the rational number
In mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...
s Q is compact and therefore locally compact in senses (1) and (2) but it is not locally compact in sense (3).
- The particular point topology
The particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then a particular point topology on X...
on any infinite set is locally compact in senses (1) and (3) but not in sense (2).
Properties
Every locally compact preregular space is, in fact, completely regular. It follows that every locally compact Hausdorff space is a
Tychonoff spaceIn topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces.These conditions are examples of separation axioms....
. Since straight regularity is a more familiar condition than either preregularity (which is usually weaker) or complete regularity (which is usually stronger), locally compact preregular spaces are normally referred to in the mathematical literature as
locally compact regular spaces. Similarly locally compact Tychonoff spaces are usually just referred to as
locally compact Hausdorff spaces.
Every locally compact Hausdorff space is a
Baire spaceIn mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
.
That is, the conclusion of the
Baire category theoremThe Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....
holds: the
interiorIn mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S". A point that is in the interior of S is an interior point of S....
of every
unionIn set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets gives a set .- Definition :A simple example:...
of countably many nowhere dense
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
s is
emptyIn mathematics, and more specifically set theory, the empty set is the unique set having no members; its size is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
.
A subspace
X of a locally compact Hausdorff space
Y is locally compact
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional combined with its reverse ; hence the name...
X can be written as the
set-theoretic differenceIn discrete mathematics and predominantly in set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation to another...
of two
closedIn topology and related branches of mathematics, a closed set is a set whose complement is open.- Equivalent definitions of a closed set :In a topological space, a set is closed if and only if it coincides with its closure...
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
s of
Y.
As a corollary, a dense subspace
X of a compact Hausdorff space
Y is locally compact if and only if
X is an open subset of
Y.
Furthermore, if a subspace
X of
any Hausdorff space
Y is locally compact, then
X still must be the difference of two closed subsets of
Y, although the converse needn't hold in this case.
Quotient spaceIn topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
s of locally compact Hausdorff spaces are
compactly generatedIn topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space X is compactly generated if it satisfies the following condition:Equivalently, one can replace closed with open in this definition...
.
Conversely, every compactly generated Hausdorff space is a quotient of some locally compact Hausdorff space.
For locally compact spaces local uniform convergence is the same as
compact convergenceIn mathematics compact convergence is a type of convergence which generalizes the idea of uniform convergence. It is associated with the compact-open topology.-Definition:...
.
The point at infinity
Since every locally compact Hausdorff space
X is Tychonoff, it can be embedded in a compact Hausdorff space b(
X) using the Stone-Čech compactification.
But in fact, there is a simpler method available in the locally compact case; the one-point compactification will embed
X in a compact Hausdorff space a(
X) with just one extra point.
(The one-point compactification can be applied to other spaces, but a(
X) will be Hausdorff
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional combined with its reverse ; hence the name...
X is locally compact and Hausdorff.)
The locally compact Hausdorff spaces can thus be characterised as the open subsets of compact Hausdorff spaces.
Intuitively, the extra point in a(
X) can be thought of as a
point at infinity.
The point at infinity should be thought of as lying outside every compact subset of
X.
Many intuitive notions about tendency towards infinity can be formulated in locally compact Hausdorff spaces using this idea.
For example, a
continuousIn topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f always contain the image of a set of points near x...
realIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
or
complexA complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
valued
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
f with domain
X is said to
vanish at infinityIn mathematics, a function on a normed vector space is said to vanish at infinity if as For example, the functiondefined on the real line vanishes at infinity.There is a generalization of this to a locally compact setting...
if, given any positive number
e, there is a compact subset
K of
X such that |
f(
x)| <
e whenever the
pointIn geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue. Thus, a point is a 0-dimensional object...
x lies outside of
K. This definition makes sense for any topological space
X. If
X is locally compact and Hausdorff, such functions are precisely those extendable to a continuous function
g on its one-point compactification a(
X) =
X ∪ {∞} where
g(∞) = 0.
The set C
0(
X) of all continuous complex-valued functions that vanish at infinity is a C* algebra. In fact, every commutative C* algebra is isomorphic to C
0(
X) for some unique (
up toIn mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e. one to which it is considered equivalent...
homeomorphismIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between two topological spaces that has a continuous inverse function...
) locally compact Hausdorff space
X. More precisely, the
categoriesIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
of locally compact Hausdorff spaces and of commutative C* algebras are dual; this is shown using the
Gelfand representationIn mathematics, the Gelfand representation in functional analysis has two related meanings:* a way of representing commutative Banach algebras as algebras of continuous functions;...
. Forming the one-point compactification a(
X) of
X corresponds under this duality to adjoining an
identity elementIn mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
to C
0(
X).
Locally compact groups
The notion of local compactness is important in the study of
topological groupIn mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology...
s mainly because every Hausdorff
locally compact groupIn mathematics, a locally compact group is a topological group G which is locally compact as a topological space. Locally compact groups are important because they have a natural measure called the Haar measure. This allows one to define integrals of functions on G.Many of the results of finite...
G carries natural measures called the
Haar measureIn mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
s which allow one to
integrateIntegration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integralis defined informally...
functions defined on
G.
Lebesgue measureIn mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration...
on the
real lineIn mathematics, the real line is the line whose points correspond to the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space...
R is a special case of this.
The Pontryagin dual of a
topological abelian groupIn mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group.That is, a TAG is both a group and a topological space, the group operations are continuous, and the group's binary operation is commutative....
A is locally compact
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional combined with its reverse ; hence the name...
A is locally compact.
More precisely, Pontryagin duality defines a self-duality of the
categoryIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
of locally compact abelian groups.
The study of locally compact abelian groups is the foundation of
harmonic analysisHarmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
, a field that has since spread to non-abelian locally compact groups.