Missing traits for S37

[Moniker1998]

Adds the missing traits for S37 except for Toronto and properties independent of ZFC.

[felixpernegger]

P16
P171
can be removed

[felixpernegger]

P115, P217 I review later

[Moniker1998]

@prabau I haven't been very careful and took @felixpernegger suggestion but it turned out to be wrong.
I rolled back to the original.

[prabau]

P85 (basically disconnected)
I kind of agree with @felixpernegger. I haven't had the time to dig into F-spaces. Wouldn't it make more sense to not add this trait for now and wait until F-space and other needed results are added to pi-base?

(unless of course a more direct argument can be found for basically disconnected = false in this case).

[prabau]

Actually, direct proof for P85: Ordinal $\omega+1$ (S20) is a clopen subset that is not basically disconnected.

[Moniker1998]

@prabau why did you close my PR?

[Moniker1998]

Just because a post mentions these properties doesn't mean it needs to be on pi-base... and that's not a reason to close my PR

[prabau]

I didn't mean to close it. Wanted to click on the Comment button and accidentally clicked on the button to its left.

Not the first time this happens to me :-)

[Moniker1998]

Oh okay.

[Moniker1998]

@prabau anything else for this PR?

[prabau]

@Moniker1998 Not sure if you saw the last part I added to https://math.stackexchange.com/questions/5108293, starting at "Remark: (generalizing the comment of Jakobian)".

You initially used the terminology "almost open map". I looked it up in the literature and found that there are conflicting usages of that name. I also found "quasi-open map" (https://en.wikipedia.org/wiki/Quasi-open_map), which seems to have the same as your intended meaning and does not seem to have much ambiguity. So I'll continue with "quasi-open map".

Then I found there is an even more general notion: "skeletal map", which also does not seem to have much ambiguity in the literature. So for maps:
open => ... => quasi-open => skeletal.

Question: Your mathse answer assumes $\varphi:X\to X_\text{Tych}$ is quasi-open. Do the results also work more generally if $\varphi$ is skeletal?

(all the examples from pi-base at least were quasi-open, and actually more. So that's not immediately relevant, but would be good to know.)

[Moniker1998]

@prabau
Yes I've seen it.

The proposition doesn't really seem to immediately generalize to skeletal maps in any way. Cozero complemented property has to do with density of certain sets so skeletal maps are fine, but here they don't seem applicable.

I suppose I haven't defined what "almost open" is so I suppose adding a wikipedia link and changing name to quasi-open would be better

[prabau]

@Moniker1998 It would be good if you could also repeat the hypothesis that the Tychonoff reflection map is quasi-open right at the top of the Proposition in the mathse answer. That would makes things immediately clear for everyone.

[Moniker1998]

@prabau what do you mean? I'm not using this assumption throughout the post