Different types of local compactness, a missing example

[Moniker1998]

Hi, I've noticed that there is no space or theorem that would give an example or counter-example to the following.

A space that's weakly locally compact, not locally compact and not locally relatively compact.

Perhaps all is needed to fill in some properties of another space

[https://topology.pi-base.org/spaces?q=Weakly+Locally+Compact+%2B+not+locally+compact+%2B+not+locally+relatively+compact](See the search result on pi-base)

[StevenClontz]

I'd look at these spaces:

https://topology.pi-base.org/spaces?q=Weakly+Locally+Compact%2B%7ELocally+Relatively+Compact

Of those, we have four where it's unknown if the space is locally compact or not:

• S42 Right Ray Topology on the Reals
• S44 Nested interval topology
• S46 Interlocking interval topology
• S49 Divisor topology

[Moniker1998]

Seems like weak local compactness is closed under disjoint union (wikipedia provides example of disjoint union of one-point compactification of Q and a particular point topology on infinite set)/ So all we need to do is take disjoint union of a space thats weakly locally compact but not locally compact, and a space thats weakly locally compact but not locally relatively compact.

[Moniker1998]

Nonetheless, maybe it would be nice to have a connected example on pi-base instead.

[Moniker1998]

S42 Right Ray Topology on the Reals

1. Weakly locally compact: If x is a point then [x-1, inf) is a compact neighbourhood of x
2. Not locally relatively compact: A closed set with non-empty interior is the whole space which is not compact That is, there is no closed compact neighbourhood
3. Locally compact: For x we can take sets [y, inf) with y < x as a neighbourhood basis of compact sets.

[prabau]

S44 Nested interval topology

Seems to me it's Alexandrov, hence locally compact.

[Moniker1998]

S46 Interlocking interval topology seems to be Alexandrov too

[prabau]

And same for S49 Divisor topology.

I agree it would be nice to have a connected example. Could be worth a question on mathse.

[prabau]

Speaking of variants of locally compact, we currently have the following. At each point:

• (P23) Weakly locally compact (WLC): there is a compact nbhd
• (P130) Locally compact (LC) = local base of compact nbhds.
• (P24) Locally relatively compact (LRC) = local base of nbhds with compact closure which is equivalent to the alias "Weakly locally closed-and-compact" (WLCC) = there is a closed compact nbhd

One property stronger than the other ones:

• "Locally closed-and-compact" = local base of closed compact nbhds.

That's in fact equivalent to [WLC + regular], and we already have theorems saying that implies the other ones. Would it be useful to introduce a name/property for that combination? I am ok with not having a separate property for this, but once in a while I have to remind myself of that equivalence. Do you guys have an opinion?

[Moniker1998]

My opinion is that I'm perfectly fine with not having the fourth definition of locally compact space, since as you say it can be described as weakly locally compact regular space.