pi-base
A locally compact non-k_2-space
Answers π-Base, Search for Locally compact+~$k_2$-space.
I added a handful of other properties as well; I'm sure there's some other relatively low-hanging fruit but didn't want to bloat this contribution.
@StevenClontz This space looks interesting. I did not take the time to take a peek until now. Will look at it over the weekend.
@prabau since Steven is not available lately, I think you could check the modifications I've added
Oops, somehow this fell off my queue. It's back in my inbox... Thanks for the feedback everyone.
@prabau since Steven is not available lately, I think you could check the modifications I've added
@Moniker1998 Sorry, I just saw this now. @StevenClontz will look at it.
I am trying to look at this PR in preview mode, but for some reason the latest version does not show up (still shows me space S210).
There's some kind of bug with the default host; try switching host to https://pi-base-bundles.s3.us-east-2.amazonaws.com for branch StevenClontz/kspacelocallycompact instead.
For the name, compare with other similar spaces:
- S150: Right "closed-ray" topology on $[0,1]\cap\mathbb Q$
- S151: Right "open-ray" topology on $[0,1]\cap\mathbb Q$
- and a few other ones
"closed-ray" was put in quotes to indicate that these are not closed in the sense of the topology, but closed in the sense of closed intervals in an ordered set.
So what would you think of the name Join of cofinite and right "closed-ray" topologies on $\omega_1+1$ ?
@StevenClontz @plp127 Actually I am wondering, is this really the join of $\tau_1$ = the cofinite topology and $\tau_2$ = the right "closed-ray" topology in the lattice of topologies of $\omega_1+1$?
The right "closed-ray" topology would have as a basis the intervals $[\alpha,\omega_1]$ for all $\alpha\in X$. In particular with $\alpha=\omega_1$, the singleton $\{\omega_1\}$ would be open, which we don't want here. The topology $\tau_2$ that we want is very close to that; we just want to exclude $\alpha=\omega_1$. (In both cases, the base for the topology is also a collection totally ordered by inclusion. So $\tau_2$ is hereditarily connected (P196).)
Suggestions? Join of cofinite and modified right "closed-ray" topologies on $\omega_1+1$ ?
And leave the details to the description itself.
@StevenClontz @plp127 Actually I am wondering, is this really the join of τ 1 = the cofinite topology and τ 2 = the right "closed-ray" topology in the lattice of topologies of ω 1 + 1 ?
The right "closed-ray" topology would have as a basis the intervals [ α , ω 1 ] for all α ∈ X . In particular with α = ω 1 , the singleton { ω 1 } would be open, which we don't want here. The topology τ 2 that we want is very close to that; we just want to exclude α = ω 1 . (In both cases, the base for the topology is also a collection totally ordered by inclusion. So τ 2 is hereditarily connected (P196).)
Suggestions? Join of cofinite and modified right "closed-ray" topologies on ω 1 + 1 ?
And leave the details to the description itself.
oh that slightly changes my mental image of the space, yeah you are correct ω_1 isn't open I guess this modified closed-ray topology could be described as the P147-coreflection of the open-ray topology? (just saying "modified" is fine too)
Interesting way to put it. P147 (P-space) coreflection is enlarging the topology so that every $G_\delta$ set becomes open, right? So things work out precisely because of the ordinals below $\omega_1$ all have countable cofinality.
What is the usual name in the literature for P147-reflection?
Interesting way to put it. P147 (P-space) coreflection is enlarging the topology so that every G δ set becomes open, right? So things work out precisely because of the ordinals below ω 1 all have countable cofinality.
What is the usual name in the literature for P147-reflection?
I don't have any literature references I just came up with that name on the spot since it's a coreflection to the full subcategory of spaces which are P147 it seemed like a not bad name.
Seems "P-space coreflection" is a thing. https://mathoverflow.net/questions/431600
Google scholar returns also papers with "P-coreflection".
But even more commonly, given a space $X$, the topology taking as a base the collection of $G_\delta$ sets of $X$ is called the $G_\delta$ topology of $X$. Many papers using it. Example: https://arxiv.org/pdf/1707.04871
Agreed that $G_\delta$ topology is a very common phrasing for a topology that declares each countable intersection of open sets to be open.
P-space (P147) = false: Is this already derived? Otherwise, it's easy to see directly.
@StevenClontz All that is there now looks good to me. Is there anything else you want to do here (https://github.com/pi-base/data/pull/1172#issuecomment-3373371747 ?) before I approve?
P-space (P147) = false: Is this already derived? Otherwise, it's easy to see directly.
Derived (https://topology.pi-base.org/spaces/S000210/properties/P000147).