Definition talk:Locally Compact Topological Group/Definition 1

I don't think so...? (Cohn says locally compact and Wikipedia agrees) The first link I find says that these notions are equivalent on Hausdorff spaces as well. (is the difference that not every compact set is the closure of an open set?) Caliburn (talk) 16:48, 4 June 2022 (UTC)

Oh yeah obviously compact sets aren't even necessarily closed when you go outside Hausdorff spaces... Caliburn (talk) 16:52, 4 June 2022 (UTC)

Does Cohn really say that each $x$ should have a neighborhood system from compact elements? I was only surprised because it is unusual for me and unnecessarily strong at that place (if there are equivalent weaker conditions, why not take the weakest one?). But if it was your intention, it is fine.--Usagiop (talk) 17:00, 4 June 2022 (UTC)

Yes, because of Housdorff, it suffices if each point has a compact neighborhood.--Usagiop (talk) 17:04, 4 June 2022 (UTC)

(edit conflict) Cohn does not say that, the second definition from Grafakos says "A topological group is called locally compact if there is an open set U containing the identity element such that $\overline U$ is compact". (they use "topological group" to mean "Hausdorff topological group") Then states that any point as a corollary. (since you can just translate the neighbourhood presumably?) Granted I've been a bit haphazard here - I'm procrastinating revising and so have been picking off definitions, I don't understand the intricacies of this definition as well as I should, but I'll get there when I write the equivalence proof and should hopefully iron it out. You're welcome to have a go at it as well, since you seem well-versed already. Caliburn (talk) 17:16, 4 June 2022 (UTC)

I just wanted to warn you that the definition of local compactness in ProofWiki is unusual one in FA. The local compactness in your book is probably called weakly local compactness in ProofWiki. These are equivalent, so no issue of correctness. But using a unnecessarily strong condition in a definition is inconvenient, as you have to verify it every time in subsequent proofs (For example, in this case, you have to verify the existence of the neighborhood system from compact elements, instead of the existence of a single compact neighborhood). Anyway everything is fine for me, this was just a side note.--Usagiop (talk) 18:26, 4 June 2022 (UTC)

Which one is the usual definition, and which is the less usual definition, are not important (depends on which book you started on, and what the state of your library is) -- the important thing is that there are different definitions for local compactness.

When using a term on $\mathsf{Pr} \infty \mathsf{fWiki}$ it is always important to check whether the definition you are linking to is actually the definition you want. If we have done our work properly, then if there is a term which has such an ambiguity depending on where you got it from, there will be an "also known as" or "also defined as" section explaining what the difficulties are. They are not just there for completism or for useful trivia, they serve a serious purpose. --prime mover (talk) 19:02, 4 June 2022 (UTC)

There is already a dedicated page Definition:Locally Compact Hausdorff Space for this concept. Linking to this page is probably the best.--Usagiop (talk) 19:17, 4 June 2022 (UTC)

Brilliant, did not know this existed. Will change. Caliburn (talk) 19:21, 4 June 2022 (UTC)