Trait Suggestion: Novak space S109 is not first countable P28

[justforphone42069-cloud]

Trait Suggestion

The space Novak Space S000109 is not first countable P000028, but this fact is not known to pi-Base today: [link to pi-Base]

Proof/References

In abstract here: https://www.sciencedirect.com/science/article/abs/pii/B9780444522085500259

[yhx-12243]

It seems derivable once #1200 is merged.

[prabau]

Yes, this will maybe be part of #1200 and #1383. But it is not clear this "Novak-Teresaka" is the same thing as Novak space S109. This will need to be checked.

[Moniker1998]

Novak space is indeed not first countable as character of Stone-Cech compactification of N at any point of the remainder is uncountable. But we probably will be adding a stronger/weaker but equally as easy to show property.

[Moniker1998]

I would try to work on it if @prabau were to review my already existing PR's for this space

[Moniker1998]

So what I think that the best property to answer instead of this one is if Novak space and $\beta\omega$ have countable tightness, first countability is false

So far I've figured out that compact spaces with countable spread have countable tightness. This will need to be added. #1538

Also both Novak space and $\beta\omega$ don't have countable spread. I'll add this one. #1537

Edit: I've found that $\beta\omega$ has tightness $\mathfrak{c}$ but the proof is non-trivial. I will add it when I read about it. As for Novak space, I hope the proof translated to that case too, then I'll make a post.

Edit2: It generalizes to Novak space too #1540