Space Suggestion: Left ray topology on $\omega_1$

[felixpernegger]

Space Suggestion

$\omega_1$ with the topology of left rays
$\{\omega_1\}\cup\{(\leftarrow,n):n<\omega_1\} = \{\emptyset,\omega_1\}\cup\{(\leftarrow,n]:n<\omega\}$.

Rationale

Gives example of not second countable + Artinian (maybe some other properties as well, havent checked in detail) as well as t0 + ~countable + Artinian.

This is also an example on a Toronto space, which we will not have many (except the obvious: finite, discrete, cofinite, indiscrete) of.

Relationship to other spaces and properties

Similar to S166 and S199. I don't think completing this space should pose any issues.

[Moniker1998]

I think that's fine to add. But maybe not right now. We can finish your PR's first, hm?

[felixpernegger]

I think that's fine to add. But maybe not right now. We can finish your PR's first, hm?

Yes, I just thought of this today, so I made an issue. I also wanna add Artinian => ~Cut point space, since it has a funny proof I think.