Editing Locally compact space (section)

===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).