Example of door space

[prabau]

I think it would be good to have as an example space the one mentioned in the answer to https://math.stackexchange.com/questions/3088994/examples-of-connected-door-spaces. It's case (3) of Theorem 1 in https://arxiv.org/abs/1809.03085 and provides another example of a space that is anticompact and not compactly generated.

What would be a good name for it? I was thinking of something like "Ultrafilter connected door space" maybe. It's not completely descriptive (as the particular point topology could be considered an ultrafilter space based on a principal ultrafilter), but it's good enough to distinguish it among other spaces in pi-base. Any other suggestions?

And we could add "door space" as a property. Not super important, but there are a few papers using that notion, so why not.

[StevenClontz]

Is this the subspace of https://topology.pi-base.org/spaces/S000111 dropping the free ultrafilter itself as a point? If so, I think they should be named similarly (and I'm not a fan of the current name for S111).

I have no problem with adding door space property as it's in the literature. Maybe "doored" as the main name as we typically use adjectives?

[StevenClontz]

Maybe S111 = "Single free ultrafilter space" (keeping "single ultrafilter topology" as an alias) and Snew = "Single free ultrafilter subspace" as main names?

[prabau]

S111 without the ultrafilter point is discrete, so it's very different.

We usually use adjectives for properties, but I think "door space" sounds much better than "doored". It is consistently used in the literature for such spaces. And pi-base has some other properties with the word "space" in them, like $G_\delta$ space, spectral space, q space, k space, M space, etc.

Talking about adjectival forms of properties, we also have "groupable", which sounds pretty horrible to me, but that would be another discussion.

[StevenClontz]

S111 without the ultrafilter point is discrete, so it's very different.

Whoops of course. Though it suggests we have two clear non-compatible ways a "single ultrafilter" can be used to define a topology. Hmm.

Point taken on the other non-adjectives, so "door space" sounds good for the contribution.

As I recall "groupable" was chosen because, like "metrizable", the topology does not induce a unique group structure, and we don't have support for modeling groups, just topologies. But we still should (A) at least add "topological group" as an alias if not revert to being the main name, and (B) just use "groupable" as rather than "groupable topology".

[prabau]

Or, following the description of P87, "homeomorphic to topological group" maybe. (would that be too long?) Maybe we can brainstorm in the meeting today.

[prabau]

Still looking for a good name. Any suggestions?

[StevenClontz]

Maybe S111 could be renamed "Single ultrafilter subspace of $\beta\omega$"?

And maybe this is "Free ultrafilter topology on $\omega$"? To distinguish from a particular point topology induced by a principal ultrafilter.

[StevenClontz]

What search would this space answer that's doesn't already have an example in pi-Base?

[prabau]

I like your name suggestions, in particular the one for S111.

One search that the new space would fulfill: π-Base, Search for door+connected+T1+multiple points

[StevenClontz]

🤦 I should have clocked that all the other connected door spaces weren't T1. Thanks! :-)