pi-base
Meta-property: Subspaces of extremally disconnected spaces
In one of my posts, I wrote up a proposition which says that subspaces of extremally disconnected space which are intersections of dense and open subspace, are extremally disconnected and $C^\ast$-embedded.
This could replace the current meta-property, since it both generalizes it, and gives explicit proof (current one references Willard without proof for one of them, the other proven in a math.stack post by Brain Scott; mine contains both proofs, the result is more general, and it includes $C^\ast$-embeddability).
My post also mentions in passing that closed subspaces of extremally disconnected spaces need not be extremally disconnected.
For referencing below:
- (1) property hereditary wrt open sets
- (2) property hereditary wrt to dense sets
- (3) property hereditary wrt intersection of open set with dense set
Please double check me on this. If $U$ is an open set and $D$ is a dense set in $X$, then $U\cap D$ is dense in $U$. So, (1) + (2) => (3). On the other hand, (3) implies each of (1) and (2). So (3) is equivalent to the conjunction of (1) and (2).
I think it's a little easier to keep (1) and (2) separate. But we could indeed add your post as a reference.
Apart from that, I need to read your post in detail. Lots of interesting things in there.
And like you said, it would be good to have the remainder $\beta\mathbb N\setminus\mathbb N$ as an example in pi-base, to show extremally disconnected is not always preserved by closed sets.
@prabau that's right, you can easily prove 3) from 1) and 2). I find it easier to say that $U\cap D$ is open in $D$, and $D$ is dense in $X$, but you can also go like you did.
Either way what I prove is both of those things separately, the advantage being that those proofs are explicit and not hidden behind an exercise.
And like you said, it would be good to have the remainder β N ∖ N as an example in pi-base, to show extremally disconnected is not always preserved by closed sets.
If you're going to do that, then keep in mind that this space is not basically disconnected, which is the actual property that we would be adding.
Also, it's not cozero complemented, a property which would follow automatically if we were to add $F$-spaces to pi-base, but maybe you don't want to do that right now, which is fine.