Editing Locally compact space (section)

==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.