Open Moniker1998 opened 5 months ago
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:
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.
Nonetheless, maybe it would be nice to have a connected example on pi-base instead.
S42 Right Ray Topology on the Reals
S44 Nested interval topology
Seems to me it's Alexandrov, hence locally compact.
S46 Interlocking interval topology seems to be Alexandrov too
And same for S49 Divisor topology.
I agree it would be nice to have a connected example. Could be worth a question on mathse.
Speaking of variants of locally compact, we currently have the following. At each point:
One property stronger than the other ones:
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?
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.
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)