Intopology and related branches ofmathematics, atopological space is calledlocally compact if, roughly speaking, each small portion of the space looks like a small portion of acompact space. More precisely, it is a topological space in which every point has a compactneighborhood.

When locally compact spaces areHausdorff they are calledlocally compact Hausdorff, which are of particular interest inmathematical analysis.^[1]

LetX be atopological space. Most commonlyX is calledlocally compact if every pointx ofX has a compactneighbourhood, i.e., there exists an open setU and a compact setK, such that${\displaystyle x\in U\subseteq K}$.

There are other common definitions: They are allequivalent ifX is aHausdorff space (or preregular). But they arenot equivalent in general:

1. every point ofX has a compactneighbourhood.

2. every point ofX has aclosed compact neighbourhood.

2′. every point ofX has arelatively compact neighbourhood.

2″. every point ofX has alocal base of relatively compact neighbourhoods.

3. every point ofX has a local base of compact neighbourhoods.

4. every point ofX has a local base of closed compact neighbourhoods.

5.X is Hausdorff and satisfies any (or equivalently, all) of the previous conditions.

Logical relations among the conditions:^[2]

• Each condition implies (1).
• Conditions (2), (2′), (2″) are equivalent.
• Neither of conditions (2), (3) implies the other.
• Condition (4) implies (2) and (3).
• Compactness implies conditions (1) and (2), but not (3) or (4).

Condition (1) is probably the most commonly used definition, since it is the least restrictive and the others are equivalent to it whenX isHausdorff. This equivalence is a consequence of the facts that compact subsets of Hausdorff spaces are closed, and closed subsets of compact spaces are compact. Spaces satisfying (1) are also calledweakly locally compact,^[3][4] as they satisfy the weakest of the conditions here.

As they are defined in terms of relatively compact sets, spaces satisfying (2), (2'), (2") can more specifically be calledlocally relatively compact.^[5][6] Steen & Seebach^[7] calls (2), (2'), (2")strongly locally compact to contrast with property (1), which they calllocally compact.

Spaces satisfying condition (4) are exactly thelocally compact regular spaces.^[8][2] Indeed, such a space is regular, as every point has a local base of closed neighbourhoods. Conversely, in a regular locally compact space suppose a point${\displaystyle x}$ has a compact neighbourhood${\displaystyle K}$. By regularity, given an arbitrary neighbourhood${\displaystyle U}$ of${\displaystyle x}$, there is a closed neighbourhood${\displaystyle V}$ of${\displaystyle x}$ contained in${\displaystyle K\cap U}$ and${\displaystyle V}$ is compact as a closed set in a compact set.

Condition (5) is used, for example, inBourbaki.^[9] Any space that is locally compact (in the sense of condition (1)) and also Hausdorff automatically satisfies all the conditions above. Since in most applications locally compact spaces are also Hausdorff, theselocally compact Hausdorff spaces will thus be the spaces that this article is primarily concerned with.

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

• theunit interval [0,1];
• theCantor set;
• theHilbert cube.

• TheEuclidean spacesRⁿ (and in particular thereal lineR) are locally compact as a consequence of theHeine–Borel theorem.
• Topological manifolds share the local properties of Euclidean spaces and are therefore also all locally compact. This even includesnonparacompact manifolds such as thelong line.
• Alldiscrete spaces are locally compact and Hausdorff (they are just thezero-dimensional manifolds). These are compact only if they are finite.
• Allopen orclosed subsets 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 spaceQ_p ofp-adic numbers is locally compact, because it ishomeomorphic to theCantor set minus one point. Thus locally compact spaces are as useful inp-adic analysis as in classicalanalysis.

As mentioned in the following section, if a Hausdorff space is locally compact, then it is also aTychonoff space. For this reason, examples of Hausdorff spaces that fail to be locally compact because they are not Tychonoff spaces can be found in the article dedicated toTychonoff spaces.But there are also examples of Tychonoff spaces that fail to be locally compact, such as:

• the spaceQ ofrational numbers (endowed with the topology fromR), since any neighborhood contains aCauchy sequence corresponding to an irrational number, which has no convergent subsequence inQ;
• the subspace${\displaystyle \{(0,0)\}\cup ((0,\mathrm{\infty })\times \mathbf{R})}$ of${\displaystyle {\mathbf{R}}^2}$, since the origin does not have a compact neighborhood;
• thelower limit topology orupper limit topology on the setR of real numbers (useful in the study ofone-sided limits);
• anyT₀, hence Hausdorff,topological vector space over${\displaystyle \mathbb{R}}$ or${\displaystyle \mathbb{C}}$ that is infinite-dimensional, 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 theHilbert 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.

• Theone-point compactification of therational numbersQ is compact and therefore locally compact in senses (1) and (2) but it is not locally compact in senses (3) or (4).
• Theparticular point topology on any infinite set is locally compact in senses (1) and (3) but not in senses (2) or (4), because the closure of any neighborhood is the entire space, which is non-compact.
• Thedisjoint union of the above two examples is locally compact in sense (1) but not in senses (2), (3) or (4).
• Theright order topology on the real line is locally compact in senses (1) and (3) but not in senses (2) or (4), because the closure of any neighborhood is the entire non-compact space.
• TheSierpiński space is locally compact in senses (1), (2) and (3), and compact as well, but it is not Hausdorff or regular (or even preregular) so it is not locally compact in senses (4) or (5). The disjoint union of countably many copies of Sierpiński space is a non-compact space which is still locally compact in senses (1), (2) and (3), but not (4) or (5).
• More generally, theexcluded point topology is locally compact in senses (1), (2) and (3), and compact, but not locally compact in senses (4) or (5).
• Thecofinite topology on an infinite set is locally compact in senses (1), (2), and (3), and compact as well, but it is not Hausdorff or regular so it is not locally compact in senses (4) or (5).
• Theindiscrete topology on a set with at least two elements is locally compact in senses (1), (2), (3), and (4), and compact as well, but it is not Hausdorff so it is not locally compact in sense (5).

• Every space with anAlexandrov topology is locally compact in senses (1) and (3).^[10]

Every locally compactpreregular space is, in fact,completely regular.^[11][12] It follows that every locally compact Hausdorff space is aTychonoff space.^[13] 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 aslocally compact regular spaces. Similarly locally compact Tychonoff spaces are usually just referred to aslocally compact Hausdorff spaces.

Every locally compact regular space, in particular every locally compact Hausdorff space, is aBaire space.^[14][15]That is, the conclusion of theBaire category theorem holds: theinterior of everycountable union ofnowhere dense subsets is empty.

AsubspaceX of a locally compact Hausdorff spaceY is locally compact if and only ifX islocally closed inY (that is,X can be written as theset-theoretic difference of two closed subsets ofY). In particular, every closed set and every open set in a locally compact Hausdorff space is locally compact. Also, as a corollary, adense subspaceX of a locally compact Hausdorff spaceY is locally compact if and only ifX is open inY. Furthermore, if a subspaceX ofany Hausdorff spaceY is locally compact, thenX still must be locally closed inY, although theconverse does not hold in general.

Without the Hausdorff hypothesis, some of these results break down with weaker notions of locally compact. Every closed set in aweakly locally compact space (= condition (1) in the definitions above) is weakly locally compact. But not every open set in a weakly locally compact space is weakly locally compact. For example, theone-point compactification${\displaystyle {\mathbb{Q}}^{\ast }}$ of the rational numbers${\displaystyle \mathbb{Q}}$ is compact, and hence weakly locally compact. But it contains${\displaystyle \mathbb{Q}}$ as an open set which is not weakly locally compact.

Quotient spaces of locally compact Hausdorff spaces arecompactly generated.Conversely, every compactly generated Hausdorff space is a quotient of some locally compact Hausdorff space.

For functions defined on a locally compact space,local uniform convergence is the same ascompact convergence.

This section explorescompactifications of locally compact spaces. Every compact space is its own compactification. So to avoid trivialities it is assumed below that the spaceX is not compact.

Since every locally compact Hausdorff spaceX is Tychonoff, it can beembedded in a compact Hausdorff space${\displaystyle b(X)}$ using theStone–Čech compactification.But in fact, there is a simpler method available in the locally compact case; theone-point compactification will embedX in a compact Hausdorff space${\displaystyle a(X)}$ with just one extra point.(The one-point compactification can be applied to other spaces, but${\displaystyle a(X)}$ will be Hausdorff if and only ifX 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${\displaystyle a(X)}$ can be thought of as apoint at infinity.The point at infinity should be thought of as lying outside every compact subset ofX.Many intuitive notions about tendency towards infinity can be formulated in locally compact Hausdorff spaces using this idea.For example, acontinuousreal orcomplex valuedfunctionf withdomainX is said tovanish at infinity if, given anypositive numbere, there is a compact subsetK ofX such that whenever thepointx lies outside ofK. This definition makes sense for any topological spaceX. IfX is locally compact and Hausdorff, such functions are precisely those extendable to a continuous functiong on its one-point compactification${\displaystyle a(X)=X\cup \{\mathrm{\infty }\}}$ where${\displaystyle g(\mathrm{\infty })=0.}$

For a locally compact Hausdorff spaceX, the set${\displaystyle C_0(X)}$ of all continuous complex-valued functions onX that vanish at infinity is a commutativeC*-algebra. In fact, every commutative C*-algebra isisomorphic to${\displaystyle C_0(X)}$ for someunique (up tohomeomorphism) locally compact Hausdorff spaceX. This is shown using theGelfand representation.

The notion of local compactness is important in the study oftopological groups mainly because every Hausdorfflocally compact groupG carries naturalmeasures called theHaar measures which allow one tointegratemeasurable functions defined onG.TheLebesgue measure on thereal line${\displaystyle \mathbb{R}}$ is a special case of this.

ThePontryagin dual of atopological abelian groupA is locally compactif and only ifA is locally compact.More precisely, Pontryagin duality defines a self-duality of thecategory 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.

• Compact group – Topological group with compact topology
• F. Riesz's theorem
• Locally compact field
• Locally compact quantum group
• Locally compact group
• σ-compact space – Type of topological space
• Core-compact space
• Core of a locally compact space

• Compactness (mathematics)
• Properties of topological spaces

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