Editing Locally compact space (section)

=== 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.