# Locally compact space

In

topology and related branches ofmathematics , atopological space is called**locally compact**if, roughly speaking, each small portion of the space looks like a small portion of acompact space .**Formal definition**Let "X" be a

topological space . The following are common definitions for "X is locally compact", and are**equivalent if "X" is a**(or preregular). They areHausdorff space **not equivalent**in general:*1. every point of "X" has a compact neighbourhood.

*2. every point of "X" has a closed compact neighbourhood.

*2‘. every point has arelatively compact neighbourhood.

*2‘‘. every point has alocal base ofrelatively compact neighbourhoods.

*3. every point of "X" has alocal 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. 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:

* theunit interval [0,1] ;

* any closedtopological manifold ;

* theCantor set ;

* theHilbert cube .**Locally compact Hausdorff spaces that are not compact***The

Euclidean space s**R**^{n}(and in particular thereal line **R**) are locally compact as a consequence of theHeine-Borel theorem .

*Topological manifold s share the local properties of Euclidean spaces and are therefore also all locally compact. This even includes nonparacompact manifolds such as the long line.

*Alldiscrete space s are locally compact and Hausdorff (they are just the zero-dimensional manifolds). These are compact only if they are finite.

*All open orclosed subset s of a locally compact Hausdorff space are locally compact in thesubspace topology . This provides several examples of locally compact subsets of Euclidean spaces, such as theunit disc (either the open or closed version).

*The space**Q**_{"p"}of "p"-adic numbers is locally compact, because it ishomeomorphic to theCantor set minus one point. Thus locally compact spaces are as useful in "p"-adic analysis as in classical analysis.**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 space ; 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**ofrational number s, since its compactsubset s all have empty interior and therefore are not neighborhoods;

* the subspace {(0,0)} union {("x","y") : "x" > 0} of**R**^{2}, since the origin does not have a compact neighborhood;

* thelower limit topology orupper limit topology on the set**R**of real numbers (useful in the study ofone-sided limit s);

* any T_{0}, hence Hausdorff,topological vector space that isinfinite -dimension al, such as an infinite-dimensionalHilbert 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.

**Non-Hausdorff examples*** The

one-point compactification of therational number s**Q**is compact and therefore locally compact in senses (1) and (2) but it is not locally compact in sense (3).

* Theparticular 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 aTychonoff 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 theBaire category theorem holds: the interior of every union of countably manynowhere 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 closedsubset 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 anopen 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 ascompact 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 Hausdorffif and only if "X" is locally compact and Hausdorff.)The locally compact Hausdorff spaces can thus be characterised as theopen subset s 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 anypositive 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 C

_{0}("X") of all continuous complex-valued functions that vanish at infinity is a C* algebra. In fact, everycommutative C* algebra isisomorphic to C_{0}("X") for someunique (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 theGelfand representation . Forming the one-point compactification a("X") of "X" corresponds under this duality to adjoining anidentity element to C_{0}("X").**Locally compact groups**The notion of local compactness is important in the study of

topological group s mainly because every Hausdorfflocally compact group "G" carries natural measures called theHaar measure s which allow one to integrate functions defined on "G".Lebesgue measure on thereal line **R**is a special case of this.The

Pontryagin dual of atopological abelian group "A" is locally compactif 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 ofharmonic analysis , a field that has since spread to non-abelian locally compact groups.**References***cite book |last = Kelley |first = John | title = General Topology |year= 1975 | publisher = Springer | id = ISBN 0-387-90125-6

*cite book | last = Munkres | first = James | year = 1999 | title = Topology | edition = 2nd ed. | publisher = Prentice Hall | id = ISBN 0-13-181629-2

*Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | origyear=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover reprint of 1978 | isbn=978-0-486-68735-3 | id=MathSciNet|id=507446 | year=1995

*cite book | last = Willard | first = Stephen | title = General Topology | publisher = Addison-Wesley | year = 1970 | id = ISBN 0-486-43479-6 (Dover edition)

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