Space Suggestion: *some* space which is not P164 (|X| < meas. cardinal)

[felixpernegger]

(P164 = Cardinality less than every measurable cardinal)

E.g. indiscrete + discrete space. Maybe there are also interesting ones.
This would fill out a huge amount of technically unknown implications, even if its not mathematically super important.

[yhx-12243]

Note: This need additional set theory axioms.

[felixpernegger]

I thought i.e. = e.g.

[Moniker1998]

I'm not in favour of adding any such space. It might mislead someone that some results are possible/implossible in ZFC

[prabau]

That reminds me of #1814. You were not in favor of adding such a space and maybe you are right. But maybe it could be useful, if we make clear what extra set-theoretic assumptions this would depend on. Just leaving this as a comment.

(Sorry, I meant to write this in #1823, but leaving this here for now)

@StevenClontz FYI

[Moniker1998]

@prabau I am in general not in favor of adding any space that relies on axioms beyond ZFC. I'd prefer pi-base to stick to ZFC, although I know spaces which rely on axioms beyond ZFC already exist here

Or, perhaps, ability to "switch off/on" spaces which depend on additional axioms

If pi-base were to implement additional measures to handle such spaces, I might be inclined for adding such space(s).

[felixpernegger]

I indeed changed my mind on this and am now convinced we should not have spaces beyond ZFC. The two existing ones ought to be removed.

Adding spaces beyond ZFC gives counterexamples to things that may not exist withozt additional axioms, which is very unfortunate.

[prabau]

Adding spaces beyond ZFC gives counterexamples to things that may not exist withozt additional axioms, which is very unfortunate.

Good point.

Bringing @ccaruvana and @StevenClontz into this discussion about the existing Luzin spaces.

[StevenClontz]

Toggling axioms should happen in the future. Until then, I'd follow a community consensus that we should only track results in all models of ZFC.

[Moniker1998]

@StevenClontz nice!