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. A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...

Formal definition

Let X be a topological space. The following are common definitions for X is locally compact, and are equivalent if X is Hausdorff (or preregular). They are not equivalent in general: Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Hausdorff may refer to: A Hausdorff space, when used as an adjective, as in the real line is Hausdorff. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

• 1. every point of X has a compact neighbourhood.
• 2. every point of X has a closed compact neighbourhood. (X is "locally closed/compact)
• 2‘. every point has a relatively compact neighbourhood. (X is "locally rel-compact")
• 2‘‘. every point has a local base of relatively compact neighbourhoods. (X is "liminally rel-compact")
• 3. every point of X has a local base of compact neighbourhoods. (X is "liminally compact")

Logical relations among the conditions: This is a glossary of some terms used in the branch of mathematics known as topology. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact. ... This article or section does not adequately cite its references or sources. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... This article or section does not adequately cite its references or sources. ...

• 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. This equivalence is a consequence of the facts that compact subsets of Hausdorff spaces are closed, and closed subsets of compact spaces are compact. Hausdorff may refer to: A Hausdorff space, when used as an adjective, as in the real line is Hausdorff. ...

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: In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...

• the unit interval [0,1];
• any closed topological manifold;
• the Cantor set;
• the Hilbert cube.

In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... In mathematics, the Hilbert cube 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.
• Topological manifolds share the local properties of Euclidean spaces and are therefore also all locally compact. This even includes nonparacompact manifolds such as the long line.
• All discrete spaces are locally compact and Hausdorff (they are just the zero-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. This provides several examples of locally compact subsets of Euclidean spaces, such as the unit disc (either the open or closed version).
• The space Q_p of p-adic numbers is locally compact for any prime number p, because it is homeomorphic to the Cantor set minus one point. Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.

Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the real line is simply the set of real numbers. ... In mathematical analysis, the Heine-Borel theorem, named after Eduard Heine and Émile Borel, states: A subset of the real numbers R is compact iff it is closed and bounded. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... 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 is a topological space analogous to the real line, but much longer. ... In topology and related fields of mathematics, 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. ... 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... 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 equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology). ... A disc of unit radius on a plane is called a unit disc. ... The p-adic number systems were first described by Kurt Hensel in 1897. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... This word should not be confused with homomorphism. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... P-adic analysis (p-adic analysis) is a branch of mathematics that deals with functions of p-adic numbers. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either 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 isn't also a Tychonoff space; there are some examples of Hausdorff spaces that aren't Tychonoff spaces in that article. But there are also examples of Tychonoff spaces that fail to be locally compact, such as: In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ...

• the space Q of rational numbers, since its compact subsets all have empty interior and therefore don't constitute neighborhoods;
• the subspace {(0,0)} union {(x,y) : x > 0} of R², since the origin doesn't 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.

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. In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... 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. ... 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. ... In mathematics, a one-sided limit is where the limit of a function is defined in moving in the positive or negative direction, but not both. ... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... :For other senses of this word, see dimension (disambiguation). ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...

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).
• The particular point topology on any infinite set is locally compact in senses (1) and (3) but not in sense (2).

In mathematics, compactification is applied to topological spaces to make them compact spaces. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... This is a topology where inclusion of a particular point defines openness. ...

Properties

Every locally compact preregular space is, in fact, completely regular. It follows that every locally compact Hausdorff space is a Tychonoff 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. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological 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 subsets is empty. In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has enough points for certain limit processes. ... 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 which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics the term countable set is used to describe the size of a set, e. ... In topology, a subset A of a topological space X is called nowhere dense if the interior of the closure of A is empty. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...

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. 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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... It has been suggested that this article or section be merged with Logical biconditional. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, the term dense has at least three different meanings. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In logic, if S is a statement of the form P implies Q, then the converse of S is a statement of the form Q implies P. In general, the verity of S says nothing about the verity of its converse. ...

Quotient spaces of locally compact Hausdorff spaces are compactly generated. Conversely, every compactly generated Hausdorff space is a quotient of some locally compact Hausdorff space. In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... In topology, a compactly generated space is a topological space X satisfying the following condition: a subspace A is closed in X if and only if A ∩ K is closed in K for all compact subspaces K ⊆ X. Equivalently, one can replace closed with open in this definition. ...

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. 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 mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ... It has been suggested that this article or section be merged with Logical biconditional. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

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; but if X is locally compact and Hausdorff, then 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). In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Partial plot of a function f. ... In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, a function on a normed vector space is said to vanish at infinity if as For example, the function defined on the real line vanishes at infinity. ... A negative number is a number that is less than zero, such as &#8722;3. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... C*-algebras are an important area of research in functional analysis. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ... Look up Up to on Wiktionary, the free dictionary 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. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ... In mathematics, the Gelfand representation in functional analysis allows a complete characterisation of commutative C*-algebras as algebras of continuous complex-valued functions. ... 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 set. ...

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. Lebesgue measure on the real line R is a special case of this. In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G &#8594; G and the inverse operation G &#8594; G are continuous maps. ... In mathematics, a measure is a function that assigns a number, e. ... 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. ... In calculus, the integral of a function is an extension of the concept of a sum. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... In mathematics, the real line is simply the set of real numbers. ...

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. In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ... In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group. ... It has been suggested that this article or section be merged with Logical biconditional. ... In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...

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. Arthur Seeback (1978). Counterexamples in Topology. 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).