S100: David Gao's ultraconnected non-contractible space

[prabau]

S100: example of a nonempty ultraconnected non-contractible space. See https://math.stackexchange.com/questions/5001004.

@david20000813 Please take a look. You can preview the changes in the dgao-ultra branch, by entering that branch in the Advanced tab of pi-base (https://topology.pi-base.org/dev).

I made the name unique enough to identify the space and hint at the main properties.

As for the various traits, you can see which ones were asserted.

In particular, T1 was not asserted but derived, and the same for compact (a kind of non-obvious derivation though). https://topology.pi-base.org/spaces/S000100/properties/P000016

Among the unknown traits, maybe some of them seem of interest to add: first-countable, second-countable, Alexandrov?

[david20000813]

Thanks! This is quite nice and the derivation for non-compactness is quite surprising. As for other properties, it is second countable since the topology is generated by a countable closed subbasis. It is not Alexandrov - I'll add this to my answer on MSE so that we can just reference that.

[david20000813]

Done - the MSE answer now includes a proof that $X$ is second countable as well as a proof that $X$ is not Alexandrov.

[prabau]

@david20000813 I have added:

• second countable = T
• Alexandrov = F
• homogeneous = F (that one was kind of trivial, so no need to refer to your post I suppose)

You can check what remains from the web preview. Maybe the only one worth adding would be Meager (true, I would think).

[david20000813]

@prabau Just proposed a small typo fix for the proof that the space is not homogeneous. Otherwise it looks great. And yes, the space is meager. The easiest argument is probably the following: $K_n$ is closed while $K_n^c$ is nonempty open. The space is hyperconnected, so if $K_n$ has interior, then $K_n \cap K_n^c$ would be nonempty, which is not true. Thus, $Kn$ is nowhere dense for all $n$. Now just observe $X = \bigcup{n \in \mathbb{Z}} K_n$.

[prabau]

I don't see your typo fix. Where?

The meager argument is worth adding to the mathse answer and we can just refer to that. If you want homogeneous to be there too, that's also ok.

[david20000813]

@prabau The typo fix shows up in my “Files changed” tab. I’ve started a review for the typo fix, I believe, since I don’t seem to be able to just make edits on my own. It’s in line 11 of the proof that $X$ is not homogeneous. It should be “$x\le y$ iff…” instead of “$x<y$ iff…”

[david20000813]

@prabau Just added meager to the MSE answer. I’m fine with keeping non-homogeneous just on pi-Base. There’s no reason to add every possible property to the MSE answer.

[prabau]

It does not show for me. Just to go through the procedure for suggesting changes, from the "Files changed" tab, did you click on the + to the left of line 11? And then something pops up to leave a comment. You can just leave the comment there. Or optionally, you can also make a "suggestion" by further clicking on the icon for "Add a suggestion".

[david20000813]

@prabau That’s a bit weird. I did exactly that, and made a suggestion by clicking on + on line 11. Let me try that again.

(Oh, the issue might be because, after I wrote the suggestion, I clicked on “start a review” instead of “add a single comment”. It should be working now, I think?)

[prabau]

@Almanzoris Would you be able to review this? (Reminder: wipe out the extended description when doing the "squash and merge")

[Almanzoris]

@Almanzoris Would you be able to review this? (Reminder: wipe out the extended description when doing the "squash and merge")

Sure! I will read it tomorrow.

[Almanzoris]

I find it good.

[Almanzoris]

@david20000813 I have noticed a small typo in the mathse answer. As it is one character, I can't edit it.

[david20000813]

@Almanzoris Thanks! Has now been fixed in the MSE answer.

[Almanzoris]

You're welcome! Good.