Hereditarily connected => alpha_i

[felixpernegger]

You can use the same argument as for order topologies which we had earlier.
This doesnt actually give any new traits, but is not in the engnine, so why not..

[prabau]

I think it would be hard for someone to guess what "use the argument above the preorder induced by the property" means exactly. Which preorder? Which property? And where above the preorder? ...

Also, do you mind explaining here how that works?

[felixpernegger]

Hm yeah probably I should write this clearer. Will do tomorrow or next week!

[felixpernegger]

I added the proof.

[prabau]

Can you please fix "use the argument above the preorder induced by the property" in your post? That sentence is grammatically incorrect and makes no sense.
Also, you have the wrong link for "hereditarily connected".

[felixpernegger]

fixed it

[felixpernegger]

Actually I think this can be generalised to Well-based => $\alpha_1$ (technically strenghening T748, but the proof of the new theorem would depends on that). I'll change my comment accordingly. This gives 4 new traits

[prabau]

@felixpernegger That's a great observation, which generalizes T748. To give it the proper importance it is due, you could write a complete separate answer to https://math.stackexchange.com/questions/5116648/do-the-arkhangelskii-alpha-i-properties-hold-for-order-topologies. And then the Remark for hereditarily connected can be made a corollary in that answer.

The result seems fine to me. The proof would show that every point $x$ is an $\alpha_1$-point. Three cases: (1) $x$ has a smallest nbhd; (2) the set of nbhds of $x$ ordered by reverse inclusion has cofinality $\aleph_0$ (or $\omega$); (3) uncountable cofinality.

Case (2) can basically redo the proof of the first countable case. So we can just replace T748 completely, not rely on it.

For case (3), it can be very simple ...
(possible inspiration at Observations 1 and 2 of https://math.stackexchange.com/a/5124727)

[prabau]

Actually, it's even a lot easier: well-based + countable => first countable.
And since for each choice of "sequences" $S_n$, etc, everything can be done in a countable subspace, which is also well-based by hereditariness, it does indeed reduce to the first countable case.

[felixpernegger]

Actually, it's even a lot easier: well-based + countable => first countable. And since for each choice of "sequences" S n , etc, everything can be done in a countable subspace, which is also well-based by hereditariness, it does indeed reduce to the first countable case.

very smart actually haha.
Just one thing, T748 is mentioned a few times in other theorems/traits (i think also on mathse), so if we indeed replace it maybe we should look out for that. Definitely doable though.

[felixpernegger]

On a similar note, we have P-space + Countable -> Alexandrov (though this is not in pi base yet), so by the same argument P-space -> \alpha_1