Traits for S140: \mathbb{R} extended by a point with cocountable open neighborhoods

[mathmaster13]

P210 may be more suitable for stack exchange; let me know.

As a bonus, this PR provides another justification for why S140 is connected through the proof of sigma-connectedness. The automatic deduction pi-base displays, which uses "strongly connected" is circular because the proof of strong connectedness relies on the assumption of connectedness. That being said, strong connectedness is still what displays as the proof, so I can just put a note in to see the proof for sigma-connectedness, if that is worthwhile.

Note that there is already a proof of X's connectedness as a corollary of our proof that X is locally connected. But it isn't linked to either, so you just see strong connectedness and assuming X is connected with no reference.

[yhx-12243]

P210 may be more suitable for stack exchange

Yes.

[mathmaster13]

P210 may be more suitable for stack exchange

Yes.

Any advice for writing SE questions? I feel like "Is $\mathbb{R}$, extended with a point with co-countable open sets, an $\alpha_1$ space?" a bit too specific of a question.

[felixpernegger]

Any advice for writing SE questions? I feel like "Is R , extended with a point with co-countable open sets, an α 1 space?" a bit too specific of a question.

"Does $\mathbb{R}$ extended by a point with cocountable neighbourhoods satisify Arkhangel'skii's $\alpha_1$ property?"

[felixpernegger]

P210 may be more suitable for stack exchange

Yes.

Any advice for writing SE questions? I feel like "Is R , extended with a point with co-countable open sets, an α 1 space?" a bit too specific of a question.

this is fine

[prabau]

Yeah, being too terse may get the question closed (= rejected).

Give a description of the space, explain why this space is interesting. Then ask about the alpha_i properties in general. Need to give a precise description of what you mean by these properties. (Usually, if the post is not reasonably self contained, it may also get closed. Maybe it's enough to refer to some other post for details in this case?)

Tell what is already known for the space, related to the alpha_i properties. Basically, need to make the question interesting.

Also, you can either ask and answer yourself. Or better, have someone else write the question and then you answer later.

[mathmaster13]

Yeah, that will get the question closed pretty quickly.

If you want this done quickly, if someone who actually knows about the alpha i properties/their motivation (I know nothing about them beyond the proof I just did, so I'd have to learn that), then they should write it. I likely will be slower to contribute this weekend.

[felixpernegger]

just copy from https://math.stackexchange.com/questions/5123855 for example

[mathmaster13]

I wrote this. Not sure if it's good. Feel free to give suggestions. I saw y'all shamelessly copy the intro from each other so I did too.

[felixpernegger]

Looks good to me, maybe consider linking https://topology.pi-base.org/spaces/S000140 when you introduce the space.

Now best just post your own answer (since you have it already) to the question, no need to overcomplicated this

[felixpernegger]

Btw, I think its a good idea to rename S140 to "$\mathbb{R}$ extended by a point with cocountable open neighborhoods$ in this PR and explicitly linking S25 in the description.

[mathmaster13]

Btw, I think its a good idea to rename S140 to "$\mathbb{R}$ extended by a point with cocountable open neighborhoods$ in this PR and explicitly linking S25 in the description.

Link S25 in S140's pibase description?

[felixpernegger]

Link S25 in S140's pibase description?

Similar as its done in S50 maybe

[mathmaster13]

Link S25 in S140's pibase description?

Similar as its done in S50 maybe

done

[mathmaster13]

P73's on the border and may also need to be SE'd. Let me know

[prabau]

@mathmaster13 Nicely written mathse question. (I still need to read the answer.)
Like suggested by Felix, can you put a link to pi-base's S140 somewhere in the question, maybe link "the following space" to it.

[mathmaster13]

@mathmaster13 Nicely written mathse question. (I still need to read the answer.) Like suggested by Felix, can you put a link to pi-base's S140 somewhere in the question, maybe link "the following space" to it.

Done. Thanks

[prabau]

Is this consistent with other spaces? I feel like I've seen both kinds of names here.

Years ago all the names were basically in English words. Then, with changes to accept LaTeX in names, we started introducing more symbols where it made sense. In particular, in this case the full name was quite unwieldy and I agree that using mathbb R makes things easier to grasp. But there are no hard and fast rules. Every case should be discussed separately if necessary.

[prabau]

P73: A style thing. No need to have {S25} everywhere. Since you already mentioned in the README that the subspace $\mathbb R$ has its usual Euclidean topology, just use $\mathbb R$ to make things more readable. (Look at the preview, "too much blue")

Seems ok to have the proof directly in pi-base, as it's not too complicated. But I want to look in more detail. Will continue tomorrow.

[prabau]

P73: it can be rewritten in a much simpler way. No need to mention closed sets. Just show that a nonempty irreducible subset must be a singleton. Intersection with $\mathbb R$ cannot have multiple points because Hausdorff ... So must be finite, and must be singleton because T1 ... Done.

[prabau]

P204: at the end of the argument you use the fact that the excluded point topology on a three-point set is path connected. Didn't you mean to use it's strongly connected instead?

[mathmaster13]

P204: at the end of the argument you use the fact that the excluded point topology on a three-point set is path connected. Didn't you mean to use it's strongly connected instead?

No. It should be "connected". We're looking at maps to the two-point space, not mathbb R, because this is about having a cut point. I changed it from path connected to connected.

[mathmaster13]

This is ready to be reviewed again

[prabau]

Sorry about not getting to this earlier. Taking another look.

[prabau]

Not clear what this is saying. We need to say that $X$ contains {S25} as a subspace with the same cardinality. But they are not isomorphic. And exhibit a specific property showing that is the case.

[mathmaster13]

Sure. I committed something for you, but it is quite blue. If you have a better idea, that's awesome.

[prabau]

The fact that they have the same cardinality is rather obvious, so we don't have to justify it. Making a suggestion below.

[prabau]

P210: for the proof that $\mathbb R$ is sequentially closed in $X$ (in https://math.stackexchange.com/questions/5126685):
For both cases 1 and 2, when it says that the sequence $\{x_i\}$ has a/no convergent subsequence, you mean a subsequence convergent in $\mathbb R$, right? May be worth specifying.
Also, for case 1, it doesn't seem that the set $F$ is necessarily closed in $\mathbb R$. I think you need to add the limit of the subsequence to it.

[mathmaster13]

P210: for the proof that R is sequentially closed in X (in https://math.stackexchange.com/questions/5126685): For both cases 1 and 2, when it says that the sequence x i has a/no convergent subsequence, you mean a subsequence convergent in R , right? May be worth specifying. Also, for case 1, it doesn't seem that the set F is necessarily closed in R . I think you need to add the limit of the subsequence to it.

Done; thanks!