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)

Maybe you could leave this definition as it is and create a new one with a similar name for the concept I described. That wouldn't be so much trouble, I guess.--Dan232 09:52, 21 August 2011 (CDT)


 * No, don't worry, I'll get round to it, it's just that it's Sunday and I have a few other demands on my resources today. I apologise for not attending to business, normal service will be resumed as and when. --prime mover 11:38, 21 August 2011 (CDT)


 * Okay, I have changed the definition, adding the old definition back as an alternative definition. We also now need to:
 * a) prove the assertion made that the definitions are equivalent in a Hausdorff space (see the redlink)
 * b) revisit the rest of the definitions / proofs dependent on this definition to ensure they are still all valid under the new correct definition.
 * I have made a start by redefining Definition:Sigma-Locally Compact but I need to tread carefully as I'm not sure of the ground I'm on yet (being completely self-taught in this area). --prime mover 01:36, 22 August 2011 (CDT)

Local Basis vs. Neighorhood Basis
There are two competing definitions of "local basis" and "neighborhood basis" here. The difference is whether the sets of the basis need to be open or not. The literature is inconsistent. is attempting to use a consistent approach, so we have defined "local basis" as being a set of open neighborhoods of $x$ such that etc., while a "neighborhood basis" is a set of neighborhoods of $x$ where the neighborhoods are not necessarily expected to be open.

Henrywen, I have reverted your edits on this, as the definition as given in Steen & Seebach specifically uses the "open neighborhoods" version of this. If you have further information on this, or a source work specifying that the neighborhoods underlying a local basis need not be open, please enter the discussion on the subject here, so we can resolve this and decide a way forward.

Many thanks. --prime mover (talk) 05:14, 25 August 2015 (UTC)