W space

[StevenClontz]

A few notes taken by myself and @jocelynbell as we worked on some ideas from topological group theory and W-spaces. In particular, adds S199 as an example of a Hausdorff countably compact topological group that is not normal.

[prabau]

Not directly relevant to your changes, but since it's about W-spaces, I'll mention something about two of the corresponding theorems.

T473 [first countable ==> W-space]: Instead of just "asserted on page 430", can we mention the diagram on page 430 together with Theorem 3.2 ?

T474 [W-space ==> Frechet Urysohn]: It's implied by the diagram on page 430, via intermediate notions like w-space, countably bi-sequential, etc. But a direct proof of it is quite simple actually. Do you think it would be worth a brief sketch?

Will give some comments tomorrow about S199.

[prabau]

I think it would be helpful to have the sigma-product $\Sigma\mathbb T^{\omega_1}$ by itself as a separate space. One could already say interesting things about that one. And then have S199 refer to it, with more interesting things.

Possible suggestion: remove S199 from this PR and add the two spaces in a separate PR.

[StevenClontz]

Possible suggestion: remove S199 from this PR and add the two spaces in a separate PR.

My preference is to keep S199 in this PR in order to answer https://topology.pi-base.org/spaces?q=%24T_2%24%2BCountably+compact%2BHas+a+group+topology%2B%7ENormal , but volunteer to add its factor spaces in a future PR and connect them appropriately. ~Then again, it doesn't answer that yet, so I need to figure out what I forgot to assert...~ Oops I had a stale copy of the data in my local storage it seems, I see it now.

[prabau]

Okay, here are some comments on S199 then.

The name for the space ("Non-normal product of two countably compact, hereditarily normal topological groups") is not good, as it does not uniquely identify it. It's more a list of some of its properties, and there will be many different spaces matching these. On the other hand, $\mathbb T^{\omega_1}\times\Sigma\mathbb T^{\omega_1}$ seems very good to me: short and no ambiguity (at the expense of no easy description in English, but that does not seem a big issue). We can add the contents of the current name as a paragraph in the text itself, to explain the interest of this space.

P21: typo: $\sigma$ --> $\Sigma$ Note: If we had $\Sigma\mathbb T^{\omega_1}$ as a separate space, we could deduce P19 instead by (compact x countably compact = countably compact).

P13: the discussion at mathse and page 5 of the paper may not be completely clear. I am curious why $\mathbb T^{\omega_1}$ is the Stone-Cech compactification of $\Sigma\mathbb T^{\omega_1}$.

And not normal is deduced from $\Sigma\mathbb T^{\omega_1}$ being not paracompact (which itself derives from ... and ...). Something about that should be mentioned here. (This is also one reason why having $\Sigma\mathbb T^{\omega_1}$ as a separate space would be helpful, allows to simplify a few things here).

[StevenClontz]

Ha, okay, you win, I'm deferring this new space to a future PR. :-)

See also conversation at https://github.com/pi-base/web/issues/159