Editing Locally compact space

{{Short description|Type of topological space in mathematics}}
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]]. More precisely, it is a topological space in which every point has a compact [[Neighbourhood (mathematics)|neighborhood]]. 

When locally compact spaces are [[Hausdorff space|Hausdorff]] they are called '''locally compact Hausdorff''', which are of particular interest in [[mathematical analysis]].{{sfn|Folland|1999|loc=Sec. 4.5|p=131}}

==Formal definition==
Let ''X'' be a [[topological space]]. Most commonly ''X'' is called '''locally compact''' if every point ''x'' of ''X'' has a compact [[neighbourhood (topology)|neighbourhood]], i.e., there exists an open set ''U'' and a compact set ''K'', such that <math>x\in U\subseteq K</math>.

There are other common definitions: They are all '''equivalent if ''X'' is a [[Hausdorff space]]''' (or preregular). But they are '''not equivalent''' in general:
:1. every point of ''X'' has a compact [[neighbourhood (topology)|neighbourhood]].
:2. every point of ''X'' has a [[closed set|closed]] compact neighbourhood.
:2′. every point of ''X'' has a [[relatively compact]] neighbourhood.
:2″. every point of ''X'' has a [[local base]] of relatively compact neighbourhoods.
:3. every point of ''X'' has a local base of compact neighbourhoods.
:4. every point of ''X'' 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:<ref name="Gompa1992">{{cite journal |last1=Gompa |first1=Raghu |title=What is "locally compact"? |journal=[[Pi Mu Epsilon Journal]] |date=Spring 1992 |volume=9 |issue=6 |pages=390–392 |url=http://www.pme-math.org/journal/issues/PMEJ.Vol.9.No.6.pdf |archive-url=https://web.archive.org/web/20150910073727/http://www.pme-math.org/journal/issues/PMEJ.Vol.9.No.6.pdf |archive-date=2015-09-10 |url-status=live |jstor=24340250}}</ref>
* 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 when ''X'' is [[Hausdorff space|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.  Spaces satisfying (1) are also called '''{{visible anchor|weakly locally compact}}''',<ref>{{cite journal |last1=Lawson |first1=J. |last2=Madison |first2=B. |title=Quotients of k-semigroups |journal=[[Semigroup Forum]] |date=1974 |volume=9 |pages=1–18 |doi=10.1007/BF02194829}}, p. 3</ref><ref>{{cite book |last1=Breuckmann |first1=Tomas |last2=Kudri |first2=Soraya |last3=Aygün |first3=Halis |title=Soft Methodology and Random Information Systems |date=2004 |publisher=Springer |pages=638–644 |chapter=About Weakly Locally Compact Spaces |doi=10.1007/978-3-540-44465-7_79|isbn=978-3-540-22264-4 }}</ref> 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 called '''locally relatively compact'''.<ref>{{citation|first=Eva |last=Lowen-Colebunders|title=On the convergence of closed and compact sets|journal=[[Pacific Journal of Mathematics]]|volume=108|issue=1|pages=133–140|year= 1983|doi=10.2140/pjm.1983.108.133 |url=https://projecteuclid.org/download/pdf_1/euclid.pjm/1102720477|mr=709705|zbl=0522.54003|s2cid=55084221 |doi-access=free}}</ref><ref>{{Cite arXiv <!-- unsupported parameter |url=https://arxiv.org/pdf/2002.05943.pdf |archive-url=https://web.archive.org/web/20220107165043/https://arxiv.org/pdf/2002.05943.pdf |archive-date=2022-01-07 |url-status=live --> |eprint=2002.05943 |last1=Bice |first1=Tristan |last2=Kubiś |first2=Wiesław |title=Wallman Duality for Semilattice Subbases |year=2020 |class=math.GN}}</ref> Steen & Seebach<ref>Steen & Seebach, p. 20</ref> calls (2), (2'), (2") '''strongly locally compact''' to contrast with property (1), which they call ''locally compact''.

Spaces satisfying condition (4) are exactly the '''{{visible anchor|locally compact regular}}''' spaces.{{sfn|Kelley|1975|loc=ch. 5, Theorem 17, p. 146}}<ref name="Gompa1992"></ref>  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 <math>x</math> has a compact neighbourhood <math>K</math>. By regularity, given an arbitrary neighbourhood <math>U</math> of <math>x</math>, there is a closed neighbourhood <math>V</math> of <math>x</math> contained in <math>K\cap U</math> and <math>V</math> is compact as a closed set in a compact set.

Condition (5) is used, for example, in [[Nicolas Bourbaki|Bourbaki]].<ref>{{cite book|last1=Bourbaki|first1=Nicolas|title=General Topology, Part I|date=1989|publisher=Springer-Verlag|location=Berlin|isbn=3-540-19374-X|edition=reprint of the 1966}}</ref>  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, these '''locally compact Hausdorff''' spaces will thus be the spaces that this article is primarily concerned with.

== 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];
* the [[Cantor set]];
* the [[Hilbert cube]].

=== Locally compact Hausdorff spaces that are not compact ===
*The [[Euclidean space]]s '''R'''<sup><var>n</var></sup> (and in particular the [[real line]] '''R''') are locally compact as a consequence of the [[Heine–Borel theorem]].
*[[Topological manifold]]s share the local properties of Euclidean spaces and are therefore also all locally compact. This even includes [[paracompact|nonparacompact]] manifolds such as the [[long line (topology)|long line]].
*All [[discrete space]]s are locally compact and Hausdorff (they are just the [[0 (number)|zero]]-dimensional manifolds). These are compact only if they are finite.
*All [[open subset|open]] or [[closed subset]]s 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'''<sub>''p''</sub> of [[p-adic number|''p''-adic numbers]] is locally compact, because it is [[homeomorphic]] to the [[Cantor set]] minus one point. Thus locally compact spaces are as useful in [[p-adic analysis|''p''-adic analysis]] as in classical [[mathematical analysis|analysis]].

=== Hausdorff spaces that are not locally compact ===
As mentioned in the following section, if a Hausdorff space is locally compact, then it is also a [[Tychonoff 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 to [[Tychonoff space|Tychonoff spaces]].
But there are also examples of Tychonoff spaces that fail to be locally compact, such as:

* the space '''Q''' of [[rational number]]s (endowed with the topology from '''R'''), since any neighborhood contains a [[Cauchy sequence]] corresponding to an irrational number, which has no convergent subsequence in '''Q''';
* the subspace <math>\{(0, 0)\} \cup ((0, \infty) \times \mathbf{R})</math> of <math>\mathbf{R}^2</math>, 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 limit]]s);
* any [[T0 space|T<sub>0</sub>]], hence Hausdorff, [[topological vector space]] that is [[Infinity|infinite]]-[[dimension]]al, 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.

===Non-Hausdorff examples===
* The [[one-point compactification]] of the [[rational number]]s '''Q''' is compact and therefore locally compact in senses (1) and (2) but it is not locally compact in senses (3) or (4).
* The [[particular 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.
* The [[disjoint union (topology)|disjoint union]] of the above two examples is locally compact in sense (1) but not in senses (2), (3) or (4).
* The [[right 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.
* The [[Sierpiński 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, the [[excluded point topology]] is locally compact in senses (1), (2) and (3), and compact, but not locally compact in senses (4) or (5).
* The [[cofinite 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).
* The [[indiscrete 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).

===General classes of examples===
* Every space with an [[Alexandrov topology]] is locally compact in senses (1) and (3).<ref>{{cite arXiv |last1=Speer |first1=Timothy |title=A Short Study of Alexandroff Spaces |eprint=0708.2136 |class=math.GN |date=16 August 2007}}Theorem 5</ref>

== 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.

== See also ==

* {{annotated link|Compact group}}
* {{annotated link|F. Riesz's theorem}}
* {{annotated link|Locally compact field}}
* {{annotated link|Locally compact quantum group}}
* {{annotated link|Locally compact group}}
* {{annotated link|σ-compact space}}
* [[Core-compact space]]

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
*{{cite book| last1=Folland | first1=Gerald B.| author-link=Gerald Folland | title=Real Analysis: Modern Techniques and Their Applications | publisher=[[Wiley (publisher)|John Wiley & Sons]] | url=https://www.wiley.com/en-us/Real+Analysis%3A+Modern+Techniques+and+Their+Applications%2C+2nd+Edition-p-9780471317166| year=1999 | edition=2nd | isbn=978-0-471-31716-6 }}
*{{cite book |last = Kelley |first = John |author-link=John L. Kelley| title = General Topology |year= 1975 | publisher = [[Springer Science+Business Media|Springer]] | isbn = 978-0387901251}}
*{{cite book | last = Munkres | first = James |author-link=James Munkres| year = 1999 | title = Topology | edition = 2nd | publisher = [[Prentice Hall]] | isbn = 978-0131816299}}
*{{Schechter Handbook of Analysis and Its Foundations}} <!--{{sfn|Schechter|1996|p=}}-->
*{{Cite book | 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]] | orig-year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446  | year=1995 }}
*{{cite book | last = Willard | first = Stephen | title = General Topology | url = https://archive.org/details/generaltopology00will_0 | url-access = registration | publisher = [[Addison-Wesley]] | year = 1970 | isbn = 978-0486434797}}
{{refend}}

[[Category:Compactness (mathematics)]]
[[Category:Properties of topological spaces]]