Has closed discrete subset of size 𝔠 (part 2)

[yhx-12243]

Continuation of #1584.

Further plan:
Part 3: Separable + P227 ⇒ ¬Countably paracompact
Part 4: S101|P227 and S1103|P227

[Moniker1998]

How we currently have it, why do we use "set" instead of "subset"?

Additionally, the second "note" here says that there are spaces for which $e(X) = \mathfrak{c}$ but which don't have a discrete closed subset of size $\mathfrak{c}$. This is not exactly true, since in ZFC+CH for example, the two are equivalent.
It's not that such spaces exist, but such spaces can exist under certain set-theoretical assumptions.

@prabau FYI

[yhx-12243]

Yes, we should change to “But there are spaces with ... under certain set-theoretical assumptions.”

[Moniker1998]

I've noticed, unless I'm wrong about this, that the only place that seems to say that we work in ZFC is the "Questions" tab.

When we say "under additional set-theoretic assumptions" (not sure if it should be set-theoretic or set-theoretical, maybe @prabau can explain that), we presume that we're talking about ZFC, but there's no indication of this.

We could say something like "such space exists in ZFC iff $\mathfrak{c}$ is a (weak) limit cardinal, which is independent of ZFC".

I am not sure how it would be good to write this. @prabau

[prabau]

Regarding "set" vs. "subset":

In the context of a given topological space $X$, some subsets are distinguished as being the "open sets" and others as being the "closed sets". One cannot talk of open set or closed set without already being in a topological space. They are of course subsets of $X$, but one normally just talks about "open set" without having to say "open subset" when the encompassing space is understood. And one routinely uses language like " $X$ has an open set such that ...", etc.
If $X$ has to be specified, one can say "open subset of $X$" or "open set in $X$". And similarly for closed sets.

Anyway, there is absolutely no ambiguity here when using "closed set". It is very common usage, and it is shorter too.

Note: the property name says "has a discrete closed set" with the words "closed" and "set" together. If one had flipped the words "discrete" and "closed", "has a closed discrete set" would not have been so good.

Thoughts?

[prabau]

I have given one suggestion, with a specific condition to illustrate. I don't think we need to be complete about it, it's just a note to mention a possible pitfall (which we all initially fell into).

About ZFC, it's an unwritten blanket assumption for the site. It should be documented somewhere in the site's wiki. Possibly also on the front page https://topology.pi-base.org/, but the need there is not as clear.
That would be a question for @StevenClontz.

[prabau]

@yhx-12243 I updated S63 directly, as I could not get it as a suggestion. Please check it.

[prabau]

@yhx-12243 what's your opinion about https://github.com/pi-base/data/pulls#issuecomment-3736722839 (closed sets vs. closed subsets)?

I have checked all the rest.

[yhx-12243]

Neutral, both are okay.

[prabau]

@Moniker1998 One pending thing from part 1.
What did you want to see for https://topology.pi-base.org/theorems/T000833 ?

[Moniker1998]

@prabau #1584 (comment)