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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

Contents 1 Formal definition 2 Examples and counterexamples 2.1 Compact Hausdorff spaces 2.2 Locally compact Hausdorff spaces that are not compact 2.3 Hausdorff spaces that are not locally compact 2.4 Non-Hausdorff examples 3 Properties 3.1 The point at infinity 3.2 Locally compact groups 4 References

Formal definition

Let X be a topological space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. The following are common definitions for X is locally compact, and are equivalent if X is a Hausdorff space (or preregular). In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space They are not equivalent in general:

• 1. every point of X has a compact neighbourhood. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space.
• 2. every point of X has a closed compact neighbourhood. In Topology and related branches of Mathematics, a closed set is a set whose complement is open.
• 2‘. every point has a relatively compact neighbourhood. In Mathematics, a relatively compact subspace (or relatively compact subset) Y of a Topological space X is a subset whose closure
• 2‘‘. every point has a local base of relatively compact neighbourhoods. In Topology and related areas of Mathematics, the neighbourhood system or neighbourhood filter \mathcal{V}(x for a point x is the In Mathematics, a relatively compact subspace (or relatively compact subset) Y of a Topological space X is a subset whose closure
• 3. every point of X has a local base of compact neighbourhoods. In Topology and related areas of Mathematics, the neighbourhood system or neighbourhood filter \mathcal{V}(x for a point x is the

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. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space 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.

Examples and counterexamples

Compact Hausdorff spaces

Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article compact space. Here we mention only:

• the unit interval [0,1];
• any closed topological manifold;
• the Cantor set;
• the Hilbert cube. In Mathematics, the unit interval is the interval, that is the set of all Real numbers x such that zero is less than or equal to x In Mathematics, a topological manifold is a Hausdorff Topological space which looks locally like Euclidean space in a sense defined below In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith 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 spaces Rⁿ (and in particular the real line R) are locally compact as a consequence of the Heine-Borel theorem. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a In the Topology of Metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states For a Subset
• Topological manifolds share the local properties of Euclidean spaces and are therefore also all locally compact. In Mathematics, a topological manifold is a Hausdorff Topological space which looks locally like Euclidean space in a sense defined below This even includes nonparacompact manifolds such as the long line. In Mathematics, a paracompact space is a Topological space in which every Open cover admits an open locally finite refinement. In Topology, the long line (or Alexandroff line) is a Topological space analogous to the Real line, but much longer
• All discrete spaces are locally compact and Hausdorff (they are just the zero-dimensional manifolds). In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " 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 Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is This provides several examples of locally compact subsets of Euclidean spaces, such as the unit disc (either the open or closed version). In Mathematics, the open unit disk around P (where P is a given point in the plane) is the set of points whose distance from P is
• The space Q_p of p-adic numbers is locally compact, because it is homeomorphic to the Cantor set minus one point. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith Thus locally compact spaces are as useful in p-adic analysis as in classical analysis. In Mathematics, p -adic analysis is a branch of Number theory that deals with the Mathematical analysis of functions of P-adic numbers Analysis has its beginnings in the rigorous formulation of Calculus.

Hausdorff spaces that are not locally compact

As mentioned in the following section, no Hausdorff space can possibly be locally compact if it isn't also a Tychonoff space; there are some examples of Hausdorff spaces that aren't Tychonoff spaces in that article. In Topology and related branches of Mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of Topological spaces But there are also examples of Tychonoff spaces that fail to be locally compact, such as:

• the space Q of rational numbers, since its compact subsets all have empty interior and therefore are not neighborhoods;
• the subspace {(0,0)} union {(x,y) : x > 0} of R², since the origin does not have a compact neighborhood;
• the lower limit topology or upper limit topology on the set R of real numbers (useful in the study of one-sided limits);
• any T₀, hence Hausdorff, topological vector space that is infinite-dimensional, such as an infinite-dimensional Hilbert space. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of Real numbers; In Mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of Real numbers; In Calculus, a one-sided limit is either of the two limits of a function f ( x) of a real variable x as x In Topology and related branches of Mathematics, the T0 spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it This article assumes some familiarity with Analytic geometry and the concept of a limit.

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 numbers Q is compact and therefore locally compact in senses (1) and (2) but it is not locally compact in sense (3). In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions
• The particular point topology on any infinite set is locally compact in senses (1) and (3) but not in sense (2). 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

Properties

Every locally compact preregular space is, in fact, completely regular. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Topology and related branches of Mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of Topological spaces It follows that every locally compact Hausdorff space is a Tychonoff space. In Topology and related branches of Mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of Topological spaces 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. In Mathematics, a Baire space is a Topological space which intuitively speaking is very large and has "enough" points for certain limit processes That is, the conclusion of the Baire category theorem holds: the interior of every union of countably many nowhere dense subsets is empty. The Baire category theorem is an important tool in General topology and Functional analysis. In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Topology, a Subset A of a Topological space X is called nowhere dense if the interior of the closure of In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members

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 subsets of Y. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is ↔ In 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 In Topology and related branches of Mathematics, a closed set is a set whose complement is open. 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. In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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. Conversion is a concept in Traditional logic referring to a "type of immediate Inference in which from a given Proposition another proposition is inferred

Quotient spaces of locally compact Hausdorff spaces are compactly generated. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying In Topology, a compactly generated space (or k -space) is a Topological space whose topology is coherent with the family of all 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. In the mathematical field of analysis, uniform convergence is a type of Convergence stronger than Pointwise convergence. In Mathematics compact convergence is a type of Convergence which generalizes the idea of Uniform convergence.

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. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group In the mathematical discipline of General topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a 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. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in

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. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, a function on a Normed vector space is said to vanish at infinity if f(x\to 0 as \|x\|\to \infty A negative number is a Number that is less than zero, such as −2 In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume 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. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. In fact, every commutative C* algebra is isomorphic to C₀(X) for some unique (up to homeomorphism) locally compact Hausdorff space X. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics and Logic, the phrase "there is one and only one " is used to indicate that exactly one object with a certain property exists In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose Topological equivalence redirects here see also Topological equivalence (dynamical systems. More precisely, the categories of locally compact Hausdorff spaces and of commutative C* algebras are dual; this is shown using the Gelfand representation. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Category theory, an abstract branch of Mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are In Mathematics, the Gelfand representation in Functional analysis (named after I Forming the one-point compactification a(X) of X corresponds under this duality to adjoining an identity element to C₀(X). In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that

Locally compact groups

The notion of local compactness is important in the study of topological groups mainly because every Hausdorff locally compact group G carries natural measures called the Haar measures which allow one to integrate functions defined on G. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In Mathematics, a locally compact group is a Topological group G which is locally compact as a Topological space. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of Locally compact topological groups and subsequently define The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Lebesgue measure on the real line R is a special case of this. In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a

The Pontryagin dual of a topological abelian group A is locally compact if and only if A is locally compact. In Mathematics, in particular in Harmonic analysis and the theory of Topological groups Pontryagin duality explains the general properties of the Fourier In Mathematics, a topological abelian group, or TAG, is a Topological group that is also an Abelian group. ↔ More precisely, Pontryagin duality defines a self-duality of the category of locally compact abelian groups. In Category theory, an abstract branch of Mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets 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. 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

References

• Kelley, John (1975). General Topology. Springer. ISBN 0-387-90125-6.  
• Munkres, James (1999). Topology, 2nd ed. , Prentice Hall. ISBN 0-13-181629-2.  
• Steen, Lynn Arthur; J. Lynn Arthur Steen is an American Mathematician who is Professor of Mathematics at St Arthur Seeback (1978). Counterexamples in Topology. Counterexamples in Topology (1970 2nd ed 1978 is a book on Mathematics by topologists Lynn Steen and J New York: Springer-Verlag. ISBN 0-486-68735-X (Dover edition).  
• Willard, Stephen (1970). General Topology. Addison-Wesley. ISBN 0-486-43479-6 (Dover edition).  

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