Editing Locally compact space (section)

== Properties ==
Every locally compact [[preregular space]] is, in fact, [[Completely regular space|completely regular]].{{sfn|Schechter|1996|loc=17.14(d), p. 460}}<ref>{{cite web |title=general topology - Locally compact preregular spaces are completely regular |url=https://math.stackexchange.com/questions/4503299 |website=Mathematics Stack Exchange}}</ref>  It follows that every locally compact Hausdorff space is a [[Tychonoff space]].{{sfn|Willard|1970|loc=theorem 19.3, p.136}}  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 regular space, in particular every locally compact Hausdorff space, is a [[Baire space]].{{sfn|Kelley|1975|loc=Theorem 34, p. 200}}{{sfn|Schechter|1996|loc=Theorem 20.18, p. 538}}
That is, the conclusion of the [[Baire category theorem]] holds: the [[interior (topology)|interior]] of every [[countable]] union of [[nowhere dense]] subsets is empty.

A [[subspace (topology)|subspace]] ''X'' of a locally compact Hausdorff space ''Y'' is locally compact if and only if ''X'' is [[locally closed]] in ''Y'' (that is, ''X'' can be written as the [[Complement (set theory)|set-theoretic difference]] of two closed subsets of ''Y'').  In particular, every closed set and every open set in a locally compact Hausdorff space is locally compact.  Also, as a corollary, a [[dense (topology)|dense]] subspace ''X'' of a locally compact Hausdorff space ''Y'' is locally compact if and only if ''X'' is open in ''Y''.  Furthermore, if a subspace ''X'' of ''any'' Hausdorff space ''Y'' is locally compact, then ''X'' still must be locally closed in ''Y'', although the [[converse (logic)|converse]] 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 a [[weakly 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, the [[one-point compactification]] <math>\Q^*</math> of the rational numbers <math>\Q</math> is compact, and hence weakly locally compact.  But it contains <math>\Q</math> as an open set which is not weakly locally compact.

[[Quotient space (topology)|Quotient space]]s of locally compact Hausdorff spaces are [[Compactly generated space|compactly 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 as [[compact convergence]].

=== The point at infinity ===
This section explores [[compactification (mathematics)|compactification]]s of locally compact spaces.  Every compact space is its own compactification.  So to avoid trivialities it is assumed below that the space ''X'' is not compact.

Since every locally compact Hausdorff space ''X'' is Tychonoff, it can be [[Embedding (topology)|embedded]] in a compact Hausdorff space <math>b(X)</math> 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 <math>a(X)</math> with just one extra point.
(The one-point compactification can be applied to other spaces, but <math>a(X)</math> 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 <math>a(X)</math> 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 function (topology)|continuous]] [[real number|real]] or [[Complex number|complex]] valued [[Function (mathematics)|function]] ''f'' with [[Domain (function)|domain]] ''X'' is said to ''[[vanish at infinity]]'' if, given any [[positive number]] ''e'', there is a compact subset ''K'' of ''X'' such that <math>|f(x)| < e</math> whenever the [[Point (geometry)|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 <math>a(X) = X \cup \{ \infty \}</math> where <math>g(\infty) = 0.</math>

=== Gelfand representation ===
For a locally compact Hausdorff space ''X,'' the set <math>C_0(X)</math> of all continuous complex-valued functions on ''X'' that vanish at infinity is a commutative [[C-star algebra|C*-algebra]]. In fact, every commutative C*-algebra is [[isomorphic]] to <math>C_0(X)</math> for some [[unique (mathematics)|unique]] ([[up to]] [[homeomorphism]]) locally compact Hausdorff space ''X''. This is shown using the [[Gelfand representation]]. 

=== Locally compact groups ===
The notion of local compactness is important in the study of [[topological group]]s mainly because every Hausdorff [[locally compact group]] ''G'' carries natural [[Measure theory|measures]] called the [[Haar measure]]s which allow one to [[integral|integrate]] [[measurable function]]s defined on ''G''.
The [[Lebesgue measure]] on the [[real line]] <math>\R</math> 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 (category theory)|duality]] of the [[category theory|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.