Definition talk:Locally Compact Space

This definition is not correct, a locally compact space is a space that has a neighbourhood basis of compact sets of each point. The definition in this page is equivalent to locally compactness only in Hausdorff spaces.Definition in nLab --Dan232 17:48, 20 August 2011 (CDT)
 * I see your source work and I'm going to suspend judgment till I've studied it. This may take some time.
 * How does strongly locally compact fit in? --prime mover 18:07, 20 August 2011 (CDT)

I've never work with "strongly locally compact" spaces, but I looked it up and this is what I found:
 * This staments are equivalent (I don't have the proof, but I believe my source)

* every point of X has a closed compact neighbourhood. * every point has a relatively compact neighbourhood. * every point has a local base of relatively compact neighbourhoods.

A relatively compact set is a set $U$ for which $U^-$ is compact. So the definition you already have for "strongly locally compact" is equivalent to that with a whole basis.

It is common to look for the definition of "locally compact" and get different answers. The reason is that, when topology was born as an mathematical area of study, all topological spaces were defined to be Hausdorff; and with Hausdorffness all those definitions were equivalent. The "correct" definition "for me" is the one I said because it follows the definition of other local properties like "local connectness". --Dan232 18:26, 20 August 2011 (CDT)


 * Sheesh. Nightmare. If this definition is wrong, it looks like we need to revisit every page linking to it to check whether that's also wrong. Bewilderingly, there's nothing in my beloved Steen & Seebach which hitherto has been fairly good at discriminating between Hausdorff and non-Hausdorff spaces.


 * The other property is $\sigma$-local compactness which might also need to be reviewed. --prime mover 02:09, 21 August 2011 (CDT)