pi-base
2-Markov Menger and cocountable subsets of reals
@StevenClontz @ccaruvana
I read some of Steven's article https://dml.cz/bitstream/handle/10338.dmlcz/146790/CommentatMathUnivCarolRetro_58-2017-2_8.pdf where he proves that a certain set-theoretic assumption implies 2-Markov Menger for Fortissimo space on $\mathbb{R}$. Same assumption works for cocountable topology on $\mathbb{R}$.
So I've asked if this space having 2-Markov Menger property is independent of ZFC:
https://mathoverflow.net/questions/492388/does-cocountable-topology-on-mathbbr-have-the-2-markov-menger-property
And if the assumption is independent of ZFC:
https://math.stackexchange.com/questions/5063919/is-mathcala-mathfrakc-independent-of-mathrmzfc
All of the properties of spaces up to S30 except for S17, S18, S22 having 2-Markov Menger property, are either done or equivalent to continuum hypothesis (as checked by me).
I realize this might be a hard problem, but I wanted to get your attention on this one. Perhaps some collaboration to solve this problem will be possible in the future, at least.
I've got a comment on my question, and it looks like @StevenClontz and Dow show in a paper from 2018 that there is a model of ZFC with $\mathfrak{c} = \aleph_2$ and $\lnot\mathcal{A}(\aleph_2)$, and so $\mathcal{A}(\mathfrak{c})$ is independent of ZFC.
I suppose it'd be useful to have gone through more than just one paper. 😅
Yeah, sorry I didn't have a chance to carefully respond and point you to this paper (in which all the set theory was due to Dow).
@StevenClontz that's okay. Some high school student used AI and pointed me to that paper (I normally don't use AI but I guess its useful for that). If you want to write a short answer, doesn't have to be careful at all, to that https://math.stackexchange.com/questions/5063919/is-mathcala-mathfrakc-independent-of-mathrmzfc question I made, feel free to - it would be nice to have an answer from the authority itself :) Otherwise it'd feel a bit wasteful of the bounty I have on it