Locally compact space
Encyclopedia
In topology
and related branches of mathematics
, a topological space
is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space
.
. The following are common definitions for X is locally compact, and are equivalent if X is a Hausdorff space
(or preregular). They are not equivalent in general:
Logical relations among the conditions:
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 Hausdorff
. 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.
.
Here we mention only:
; 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 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.
. 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 space
.
That is, the conclusion of the Baire category theorem
holds: the interior
of every union
of countably many nowhere dense subset
s is empty
.
A subspace X of a locally compact Hausdorff space Y is locally compact if and only if
X can be written as the set-theoretic difference
of two closed
subset
s of Y.
As a corollary, a dense subspace X of a locally 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 space
s of locally compact Hausdorff spaces are compactly generated
.
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 convergence
.
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 if
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 continuous real
or complex
valued function
f with domain X is said to vanish at infinity if, given any positive number e, there is a compact subset K of X such that |f(x)| < e whenever the point
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 C0(X) of all continuous complex-valued functions that vanish at infinity is a C* algebra. In fact, every commutative C* algebra is isomorphic to C0(X) for some unique (up to
homeomorphism
) locally compact Hausdorff space X. More precisely, the categories
of locally compact Hausdorff spaces and of commutative C* algebras are dual; this is shown using the Gelfand representation
. Forming the one-point compactification a(X) of X corresponds under this duality to adjoining an identity element
to C0(X).
s mainly because every Hausdorff locally compact group
G carries natural measures called the Haar measure
s which allow one to integrate
functions defined on G.
The Lebesgue measure
on the real line
R is a special case of this.
The Pontryagin dual of a topological abelian group
A is locally compact if and only if
A is locally compact.
More precisely, Pontryagin duality defines a self-duality of the category
of locally compact abelian groups.
The study of locally compact abelian groups is the foundation of harmonic analysis
, a field that has since spread to non-abelian locally compact groups.
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...
and related branches of mathematics
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...
, 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...
is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space
Compact 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...
.
Formal definition
Let X be a topological spaceTopological 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...
. The following are common definitions for X is locally compact, and are equivalent if X is a Hausdorff space
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...
(or preregular). They are not equivalent in general:
- 1. every point of X has a compact neighbourhood.
- 2. every point of X has a closedClosed setIn 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...
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 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...
. 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 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...
.
Here we mention only:
- the unit intervalUnit intervalIn mathematics, the unit interval is the closed interval , 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 manifoldTopological manifoldIn mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
; - the Cantor setCantor setIn mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
; - the Hilbert cubeHilbert cubeIn 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 spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
s Rn (and in particular the real lineReal lineIn mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R) are locally compact as a consequence of the Heine-Borel theorem. - Topological manifoldTopological manifoldIn 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 lineLong line (topology)In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore it serves as one of the basic counterexamples of topology...
. - All discrete spaceDiscrete spaceIn 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 (number)0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 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 topologySubspace topologyIn 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 S of X, the...
. 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 numbersP-adic numberIn mathematics, and chiefly number theory, the p-adic number system for any prime number p 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 setCantor setIn mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
minus one point. Thus locally compact spaces are as useful in p-adic analysisP-adic analysisIn mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers....
as in classical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
.
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 spaceTychonoff space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are 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 numberRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s (endowed with the topology from R), since its compact subsetSubsetIn 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 all have empty interiorInterior (topology)In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary 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)} unionUnion (set theory)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 S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
{(x,y) : x > 0} of R2, since the origin does not have a compact neighborhood; - the lower limit topologyLower limit topologyIn 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 limitOne-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 spaceTopological vector spaceIn mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
that is infinite-dimensionDimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any 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 spaceHilbert 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 numberRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
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 topologyParticular point topologyIn mathematics, 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 the particular point topology...
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 spaceTychonoff space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are 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 space
Baire space
In 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 theorem
Baire category theorem
The 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 interior
Interior (topology)
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
of every union
Union (set theory)
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 S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of countably many nowhere dense 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 is empty
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality 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 if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
X can be written as the set-theoretic difference
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of two closed
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...
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 Y.
As a corollary, a dense subspace X of a locally 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
Converse (logic)
In logic, the converse of a categorical or implicational statement is the result of reversing its two parts. For the implication P → Q, the converse is Q → P. For the categorical proposition All S is P, the converse is All P is S. In neither case does the converse necessarily follow from...
needn't hold in this case.
Quotient space
Quotient space
In 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 generated
Compactly generated space
In 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 convergence
Compact convergence
In 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 if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
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 continuous real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
or complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
valued function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f with domain X is said to vanish at infinity if, given any positive number e, there is a compact subset K of X such that |f(x)| < e whenever the point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
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 C0(X) of all continuous complex-valued functions that vanish at infinity is a C* algebra. In fact, every commutative C* algebra is isomorphic to C0(X) for some unique (up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
) locally compact Hausdorff space X. More precisely, the categories
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
of locally compact Hausdorff spaces and of commutative C* algebras are dual; this is shown using the Gelfand representation
Gelfand representation
In 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 element
Identity element
In 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 C0(X).
Locally compact groups
The notion of local compactness is important in the study of topological groupTopological group
In 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. A topological group is a mathematical object with both an algebraic structure and a...
s mainly because every Hausdorff locally compact group
Locally compact group
In 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 measure
Haar measure
In 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 integrate
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
functions defined on G.
The Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
on the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R is a special case of this.
The Pontryagin dual of a topological abelian group
Topological abelian group
In 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 if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
A is locally compact.
More precisely, Pontryagin duality defines a self-duality of the category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
of locally compact abelian groups.
The study of locally compact abelian groups is the foundation of harmonic analysis
Harmonic analysis
Harmonic 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.