Scattered sequentially discrete spaces are anticompact

[yhx-12243]

I'm not sure whether 𝑘₁-Hausdorff can be dropped, since I have not discovered any counterexamples.

See #1549 (comment). A helper theorem adapted from Corollary 3.5.

[Moniker1998]

What does it mean for $x\in K$ to be Cantor-Brendixson rank $1$?

[yhx-12243]

https://en.wikipedia.org/wiki/Derived_set_(mathematics)#Cantor%E2%80%93Bendixson_rank

Equivalently, let 𝑥 ∈ 𝑋′ ∖ 𝑋″.

[Moniker1998]

@yhx-12243 that's Cantor-Brendixson rank for a set. So does this notion for a point of a set exist anywhere?

[yhx-12243]

https://math.stackexchange.com/questions/4922798/equivalent-definitions-of-cantor-bendixson-rank

rank(𝑝) = 𝛼 iff 𝑝 ∈ 𝑋⁽ᵅ⁾ ∖ 𝑋⁽ᵅ⁺¹⁾.

[Moniker1998]

@prabau about P130 here, do you like how it's written or would you prefer it separated into three points like for P49?

[Moniker1998]

I'm not sure whether 𝑘₁-Hausdorff can be dropped, since I have not discovered any counterexamples.

As for this I think it'd be somewhat safe to assume that it can't be dropped since it does somewhat heavy lifting in this proof.
But maybe it could be weakened. Either way, this is something we probably shouldn't worry about.

[prabau]

@prabau about P130 here, do you like how it's written or would you prefer it separated into three points like for P49?

I think it's good separated out as you have it now.

[prabau]

I just read the proof, and I can make sense of it, but some steps don't flow as well as they could.
Let me see if I can suggest some minor tweaks.

[prabau]

One thing in the current proof is that one needs to keep track of various nbhds, and one needs to know if we are talking of nbhds in $X$ or nbhds in $K$, which is not always clear, and annoying.

We need to show that $K$ is finite.
But all the properties in the hypotheses are hereditary (I'll add it for k1-Hausdorff) and still hold for $K$. So without loss of generality, we can restrict attention to $K$ and assume $X$ itself if compact Hausdorff and show $X$ is finite.

Would it be ok with you if I suggest something along these lines to simplify things?

[prabau]

I added a suggestion, which you need to look at in preview mode and compare with the previous version to see if it makes sense. It ended up not being shorter, but I explained in more detail some things that were not immediately clear to me before.

Would that be too long?

[yhx-12243]

I added a suggestion, which you need to look at in preview mode and compare with the previous version to see if it makes sense. It ended up not being shorter, but I explained in more detail some things that were not immediately clear to me before.

It's okay.

Would that be too long?

Maybe a little bit, I'm not sure.

[prabau]

btw, does this theorem help derive some new traits?

[yhx-12243]

Originally it helps for #1549.

I've experienced that it also helps for S181 (Countable σ-product of ω + 1) if one add S181|P51. However, this space are very under development.