Property Suggestion: σ-closed compact

[Moniker1998]

Property Suggestion

A space is said to be $\sigma$-closed compact if $X = \bigcup_n K_n$ where $K_n$ are closed and compact

Rationale

This property arised in the discussion about distinguishing different properties based on relative compactness. It does not seem to be equivalent to any combination of properties on pi-base, and refines some theorems.

Relationship to other properties

1. $\sigma$-closed compact implies metacompact and $\sigma$-compact
2. locally relatively compact and $\sigma$-compact implies $\sigma$-closed compact
3. KC and $\sigma$-compact implies $\sigma$-closed compact

[prabau]

I am not completely convinced this property is notable enough. It could be, but I'd like to think more about it and discuss in the other thread #1684.

Will add something to the general discussion about $\sigma$-(some form of relatively compact set) in the other thread.

[Moniker1998]

@prabau notable? Also I've already discussed $\sigma$-(some form of relatively compact set). Did you read my messages?

I don't really care if something is notable for anyone. That alone doesn't matter.

[Moniker1998]

There is a lot of little properties like that we care about solely because they matter for spaces which aren't Hausdorff. This is one of them. Think of it as a version of "locally relatively compact" but a version for $\sigma$-compactness.

[prabau]

To have everything self-contained here, I'll summarize the discussion starting in this thread #1684 (comment).

Following Definitions of relatively compact there are essentially three non-equivalent definitions of a subset A of a topological space X being "relatively compact". For the sake of this discussion, let's call them

• RC1: A is contained in a closed compact subset of X
• RC2: A is contained in a compact subset of X
• RC3: definition based on open covers of X

To each of these definitions corresponds a definition for X being a countable union of subsets, each being RC-i in X:

• $\sigma$-RC1 = $\sigma$-closed compact, as proposed in this PR.
• $\sigma$-RC2 = $\sigma$-compact
• $\sigma$-RC3 = current property P71 (called $\sigma$-relatively compact), equivalent to P70 (Markov Menger)

Now, $\sigma$-RC2 and $\sigma$-RC3 do appear in the literature. Apparently $\sigma$-RC1 does not as a separate topic of study (various reasons, including the fact that for Hausdorff spaces (or KC spaces) RC1 = RC2, and for regular spaces RC1 = RC2 = RC3).

I do agree though that it could be interesting to have to show subtle differences between properties. Since $\sigma$-RC1 => $\sigma$-RC2 => $\sigma$-RC3, what are examples of spaces showing that these implications do not reverse?

One question, which @Moniker1998 and I cannot decide alone: Is the property "notable enough" for pi-base, given that there is no literature on it?

FYI @StevenClontz and others

[Moniker1998]

@prabau are the different versions of local compactness notable enough? Personally, I don't think so.

Your metric is not good enough for this issue.

[prabau]

Notable would be: has it appeared in the literature? Is is used in the literature under other names or even without a name?
Are there theorems relating this to other properties? You gave a few.
Are there good examples to distinguish from similar properties? Still need this.

(Note: we should not confuse "relatively compact" (which is a property of subsets of a space) with these "sigma" versions (which are properties of the whole space). Here we are only discussing the sigma properties.)

I am not personally opposed to the new property. But @StevenClontz (and others) should tell us more about what constitute a notable property worth having in pi-base.

[Moniker1998]

@prabau so is the issue that you aren't sure if there exist spaces which are $\sigma$-compact but not $\sigma$-closed compact?
https://topology.pi-base.org/spaces?q=sigma-compact+%2B+not+metacompact
because this is easily searchable

[Moniker1998]

@prabau I also find it weird that you introduced the notion of a "notable" property, yet at the same time you ask @StevenClontz to define it. That strikes me as strange.

[StevenClontz]

I'm sure @prabau has an idea of what he considers notable; we just need to be sure there's consensus.

[StevenClontz]

Let's look back at

https://github.com/pi-base/data/wiki/Reviewing#requirements-for-a-new-property (*)

Effectively, when we say "notable", I think we mean "of interest to researchers, whether it appears in the peer-reviewed literature or another scholarly venue such as MathOverflow". So it's not disqualifying if it's not in a peer-reviewed paper, but what I would prefer to see is a reference to discussion of this property by a broader audience than just pi-Base contributors.

(*) There are a few things in our wiki that I wonder if we should put in the repository itself, so that they are subject to the same peer review and discussion as other contributions, but this is off-topic.

[prabau]

Yes, that's what I understood by "notable" (i.e., notable enough to be added to pi-base), which by the way is not something I came up with. I remember this coming up in previous discussion as well (e.g., #547 (comment)) and in old zoom discussions.

In this case, the property does not seem to match the criteria above (mainly because for Hausdorff spaces or regular spaces there is no difference with $\sigma$-compact, and most people don't care much about non-Hausdorff non-regular spaces).

But after reexamining the proposal at the top, it seems to answer a legitimate question one could have: $\sigma$-compact = countable union of compact sets; is that equivalent to being a countable union of closed compact spaces?
The various related theorems listed at the top also seem of interest. And there are good example/counterexample spaces.
And the "notability" would be at a similar level as P166 (Has a coarser separable metrizable topology) or P233 (Has open path components) for example.

So I would be in favor of making an exception and adding this property.

@StevenClontz @felixpernegger @yhx-12243 others, opinions?

[StevenClontz]

FWIW P166 was in a book: https://doi.org/10.1007/BFb0098389
and with T_(3.5) has applications in C_p theory. (Actually, I think that probably the result is true even without $T_{3.5}$ since I believe the function space induces a coarser $T_{3.5}$ topology with a coarser separable metrizable topology.)

Anyway I cut the knot and posted https://math.stackexchange.com/questions/5130486 which might reveal how interesting this idea is to other people. (I hinted S8 hard but didn't answer it myself.)