# 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 Space 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)

- Okay, I have changed the definition, adding the old definition back as an alternative definition. We also now need to:

## Contents

## 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. $\mathsf{Pr} \infty \mathsf{fWiki}$ 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)

## Avoid the issue by defining Locally Compact Hausdorff Space?

Given that:

- There are many non-equivalent definitions of Definition:Locally Compact Space
- It is hopeless to reconcile all of them with what is found in literature
- The only option would be to give a different name to each of them
**But**all of those definitions are equivalent in Hausdorff spaces

I suggest to create a separate definition page for Locally Compact Hausdorff Space:

- This allows to continue writing about Locally Compact Hausdorff spaces, which quite often appears as a condition in theorems.
- The battle about Definition:Locally Compact Space can then be fought without harming all existing work.

What do you think? --barto (talk) 16:06, 11 March 2017 (EST)

- This sounds like a valid approach. We could even add explanatory remarks on this page, addressing these problems.
- I would suggest adding definitions based on all the definitions locally compact that we have come up with a reasonable name for. Or which we have decided an other viable approach for.
- Good suggestion! — Lord_Farin (talk) 13:33, 13 March 2017 (EDT)

- Seconded. It would be something to aim for if we could somehow pull all the threads together and explain all the different approaches and what implications we have. Then we would have created what may well be a truly useful resource.

- I recommend we open a new category "Local Compactness" with its associated Definition category, so we gather everything together under the same roof. --prime mover (talk) 14:50, 13 March 2017 (EDT)

- Agreed. Note that the category Category:Locally Compact Spaces exists already. --barto (talk) 17:44, 13 March 2017 (EDT)

- This has now been done. --prime mover (talk) 19:36, 13 March 2017 (EDT)

## Disambiguating using "Weakly locally compact"

I've seen **weakly locally compact** being used for spaces where every point has (at least) one compact neighborhood: https://link.springer.com/chapter/10.1007/978-3-540-44465-7_79 and http://ejpam.com/index.php/ejpam/article/viewFile/2199/430, definition 2. (Other sources define weakly locally compact as what on proofwiki is called strongly locally compact: https://math.stu.edu.cn/upload/LWCG/nonmetric3publisher.pdf page 115, which is strange, since it is weaker than locally compact (LC for short) in the sense below (Hausdorff space or not). Either way, I think existing literature about it should only be used as a guide; see below.)

I've seen a lot of discussions, some of which from a category-theory point of view, concluding that

*Every point has a neighborhood basis of compact sets*

is the most useful definition of LC, not only because it gives rise to the intuitive "X is LC => every neighborhood of a point is LC" reflecting the word "locally", but also for deep reasons which, frankly, I am not experienced enough for to understand.

I am convinced this is an issue where (even today's) sources contradict each other so much that literature should only be used as a *guide* to make decisions. Chances are ProofWiki will be the unique source to fully cover the issue, so I think it's no problem if ProofWiki takes the liberty to decide what is the best terminology. Since every definition that is in use is at least as strong as what I called "Weakly LC" above, it seems safe to start using "Weakly LC" as a first step in the disambiguation process. What do you think?

Note: Steen&Seebach defines LC as the "Weakly LC" above. --barto (talk) 10:05, 2 April 2017 (EDT)

- I'm a little wary of "existing literature about it should only be used as a guide" unless we have a really solid logical underpinning developed.

- If the definition in S&S is different from what it says on this page (I haven't checked, I'm not in the right headspace to consider it at the moment) then we need to either change the page, or move the citation onto a page which
*does*match what S&S says. This page was written before such rigorous correlation was implemented, and it was partly as a result of the fact that it was proving difficult to get a unified consensus of exactly what "the" definition of a concept was, that the idea of linking every such definition back to a published source was initiated. --prime mover (talk) 10:14, 2 April 2017 (EDT)

## done

All pages linking to this have been reviewed.

- If they were from Steen&Seebach, "Locally Compact" has been replaced by "Weakly Locally Compact", except for theorems about concrete examples of spaces that are Hausdorff.
- Some contain mistakes due to various confusions. I left those pages as they were and asked there for explanation.
- No change has been made to Definition:Sigma-Locally Compact Space

--barto (talk) 14:57, 9 September 2017 (EDT)

- A commendable effort. Thank you. — Lord_Farin (talk) 16:09, 13 September 2017 (EDT)