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Merge P70 (Markov Menger) and P71 (sigma-relatively compact) #1684
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- changed the title
[-]Merge P70 and P71[/-][+]Merge P70 (Markov Menger) and P71 ($sigma$-relatively compact)[/+]on Mar 19, 2026 - changed the title
[-]Merge P70 (Markov Menger) and P71 ($sigma$-relatively compact)[/-][+]Merge P70 (Markov Menger) and P71 (sigma-relatively compact)[/+]on Mar 19, 2026 See the conversation in #340 for an earlier discussion on the same topic.
About P71 (
$\sigma$ -relatively compact), I have always found the text horribly confusing. It uses the terminology of a set being "relatively compact to"$X$ , which is not quite compatible with other uses of "relatively compact". This creates no end of confusion. See the long thread starting at #1132 (comment) until the end of that PR. @StevenClontz gives some context for his terminology at the end and eventually had to admit: "As long as the ambient space X is clear from context, "relatively compact", "relatively compact to X", and "relatively compact in X" all mean the same thing to me."(#1132 (comment)) There needs to be a consistent use of the "relatively compact" terminology in pi-base. We were going to open a separate discussion to try to resolve some of that confusion, but it fell by the wayside.
So basically, independently of merging the two properties or not, P71 is not good as written. If we get rid of it, we can incorporate some of it in a modified form into P70. Would need to decide maybe a new name, or a description with a disclaimer that it conflicts with other uses of the term. Or merge P70 into P71, but P71 by itself is problematic as written.
@yhx-12243 @felixpernegger @Moniker1998 and others: Any thoughts about the "relatively compact" mess?
@StevenClontz and @ccaruvana: We would really like your input on this.
Apart from all that, I also agree with Felix on the need for #1643 to improve the properties related to selection principles and topological games.
Reacted by yhx-12243@prabau are we arguing about semantics here? If so, sorry for the harsh wording, but I don't really care either way.
Hmm, not sure what you mean by semantics. It's real concerns, but ok if you don't have an opinion.
@prabau Maybe semantics is a wrong word, I mean "relatively compact", "relatively compact to X", and "relatively compact in X"
That was just a symptom of poor exposition in pi-base, from the discussion in #1132. The main point is that relatively compact is used incompatibly in different parts of pi-base (and in some papers), and without any cautionary note.
And that should be changed.
I agree the point that the name “relatively compact” should be considered again (maybe rephrase).
Again, I strongly prefer to merge P70 and P71.
Again, I strongly prefer to merge P70 and P71.
@yhx-12243 Which main name would you suggest (i.e., add P71 to P70, or the reverse)?
It can be discussed then, basically no opinion about that.
(maybe a little bit P71, reason is that this is a pure topological property)I understand the appeal of P71 (more purely topological). Unfortunately it uses incompatible terminology with the usual meanings of "relatively compact" in pi-base and elsewhere. In that sense, P70 may be preferable, and P70 also has more obvious relations with other selection/game related properties.
@StevenClontz @ccaruvana please help.
Reacted by yhx-12243Make sense.
2 remaining items
@prabau can you explain the problem clearly?
@Moniker1998 The MSE post Definitions of relatively compact and answers show that there are essentially three non-equivalent definitions of a "relatively compact" subset of a topological space in general.
Pi-base is confusing/sloppy because it uses that same terminology with different meanings in different places. This should be corrected somehow.
@prabau then my suggestion would be to merge the properties, creating
$\sigma$ -relatively compact as an alias.
Here is a discussion and reason why I think this is a good idea:
The property using definition 2 already exists: it's$\sigma$ -compact.
The property using definition 1 does not exist (it seems), it's equivalent that$X$ is a countable union of closed compact subsets. Might be worth investigating.About the new definition, which I'll call
$\sigma$ -closed compact (union of countably many closed compact sets).We have the following implications, though I haven't found an equivalent condition with properties known to pi-base.
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$\sigma$ -closed compact implies$\sigma$ -compact & metacompact -
$\sigma$ -compact & KC implies$\sigma$ -closed compact -
$\sigma$ -compact & locally relatively compact implies$\sigma$ -closed compact
This will refine some of the theorems.
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I support merging P70 and P71.
I was curious how old P72 2-Markov Menger was, and wasn't surprised that it was something I added in graduate school when pi-Base contributions didn't require peer review: 39ef102 is the commit where this property was ported to the Git repository we use today. While it's something I personally thought about earlier in my career, I don't think it meets the notability we expect for pi-Base today.
Limited information strategies in selection games is somewhat peculuar to my research (particularly earlier in my career as it was my PhD topic). I think
$\sigma$ -relatively compact (or something like it) is the better choice for a canonical name for this property in pi-Base. But I don't have a strong opinion. I'd also support the cleaning up of what "relatively compact" or "relatively P" means consistently in pi-Base, but maybe we'd have the same problem where you can't do that without having a name that doesn't match the literature for some P, e.g. locally simply connected.Another observation: locally P is a topological property for each topological property P. But "relatively compact" as in P70/P71 is not a topological property -- it's a property of subspaces relative to a given space.
FYI, chapter b-02 in Encyclopedia of General Topology is dedicated to such "relative properties" for subspaces relative to a given space. In particular, for a subset Y of a space X the usual terminology seems to be "Y is compact in X", and not Y is "relatively compact". (And similarly for other relative properties of Y wrt X).
(An observation of how Steven's terminology came about and does make some sense)
Following up on the usual terminology for these "relative properties" based on chapter b-02 in Encycl. of Gen. Top, i.e., properties of a subspace/subset Y relative to a given space X:For Def 3 in https://math.stackexchange.com/questions/4702452, the usual terminology in the literature is not "Y is relatively compact in X". It's "Y is compact in X"; in other words "Y is compact relative to X" (my interpretation). And from there it's a very small leap to get to the confusing "Y is relatively compact in X" ( or "to X").
In that sense, if X is a countable union of subsets Y that are compact relative to X, it seems natural one could describe X as being a countable union of relatively compact subsets in the above sense. I.e., "X is
$\sigma$ -relatively compact".
So that explains to me the terminology chosen by Steven in his paper.
(The drawback is that it conflicts with other well-established uses of the same terminology.)(Defending Steven from a decade ago:) That chapter is called "relative properties" and the first sentence of section 3 "relative compactness" is "... we shall introduce the relative compactness type properties", and the chapter uses the phrase "relatively P" pretty much interchangably. 😅 If there are significant papers studying "relatively compact" as in "has compact closure" that don't also assume separation axioms that make it equivalent to "compact in", I haven't read them.
But in 2026 I don't actually care that much what color we paint this bikeshed, if the community has an alternative consensus on what should be the canonical name here. 🙃
Reacted by Patrick RabauGiven that we want to merge the two properties, one issue remains.
What should the main name for the combined property be: "Markov Menger" (P70) or "$\sigma$-relatively compact" (P71) (or some other variant)?"Markov Menger" (P70)
pros:- fits well in the grid of other game/selection principle related properties
- more important in the literature than P71
cons:
- requires more background about games/selection principles, so less "purely topological" in a way
"$\sigma$-relatively compact" (P71)
pros:- requires less background to explain, more "purely topological"
cons:
- does not seem to appear in the literature, except in one place as equivalent to Markov Menger
- confusing name, incompatible with usual uses of "relatively compact"
There is also "Markov
$\Omega$ -Menger", currently part of P70 and non-trivially equivalent to it. I don't think anyone would want it as a main name, but it's there to illustrate different concepts turning out to be equivalent.If we choose P70 as the main name, the description of P71 that will appear in the text of the combined page can be changed to not use he confusing terminology, but explain in more detail what is meant exactly.
@StevenClontz @ccaruvana @felixpernegger @yhx-12243 @Moniker1998 and others: thoughts, preference?
I really dont think this is very important; "both" properties are obviously quite obscure anyways.
I would prefer P71, simply because P70 is not really defined right now in a meaningful way.
If game theory properties get overhauled things might change.I don't think it matters if the definition of Markov Menger refers to some external source at the moment. It is well defined, but not within pi-base. Fixing that is a separate issue. We can still incorporate P71 into P70.
@prabau should we merge Markov
$\Omega$ -Menger too?
I know this has been discussed before (I see #910), but it makes no sense at all to have two equivalent properties. On the outside they may seem very different, but we have this for a couple of properties. The theorems can be easily merged (just adding a small note what definition one is using).
If one wanted to introduce a new property today that looks different, but is in fact equivalent to a property we already have, there is no way we would accept it. So why have a different standard for those properties? Especially since "Markov Menger" is used in pretty much 1 paper in total...
On another note, as I said in #1643, the game theoretic properties are written down very poorly and unfortunately I got no reaction on that issue.
Several of those properties do not actually appear in the literature (except maybe 1 paper) and for example T168 has no counterexamples (and neither of the properties involved in that theorem are properly stated).
Again, if one were to suggest adding these properties today they would probably not be accepted. (Unlike say Artinian, which also doesnt have many references, I dont think they carry super interesting properties)
So personally, if the status of those properties is not improved soon, I would be in favour of removing at least some of them (like 2-Markov-Menger), I just dont really see a good reason to keep them.