Editing Locally compact space (section)

== 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>