Work on $\omega_1$ (S35)

[felixpernegger]

Once #1556 and #1455 and this PR are merged, we only need

• Proximal (P76)
• Strongly collectionwise normal (P207), see Theorem Suggestion: Two theorems about GO-spaces and ordinal spaces #1568.
• Embeds in a topological $W$-group (P186)

for this space (S35) to complete all ordinal spaces.

[yhx-12243]

P76: See Theorem 6.2 in {{zb:1375.54007}}.

[felixpernegger]

Thanks, I included it in the PR. How did you find this by the way?

[Moniker1998]

That $\omega_1$ embeds into a $W$-group is a corollary of proposition 7.2 in Products of normal spaces by Przymusiński in Handbook of set-theoretic topology by Kunen and Vaughan. It embeds into a uncountable $\Sigma$-product of $\mathbb{R}$, so that into a $W$-group.

[Moniker1998]

In basic language, let $f:\omega_1+1\to \{0, 1\}^{\omega_1}$ be defined as $f(\alpha) = x^\alpha$ such that $x^\alpha_\beta = 1$ iff $\beta\in \alpha$. Then it's easy to check that $f$ is an embedding, simply take preimage of $\{x : x_\beta = 1\}$ and notice how it's clopen in $\omega_1+1$. So this is an embedding. And its restriction to $\omega_1$ is an embedding into the $\Sigma$-product of $\omega_1$ copies of $\{0, 1\}$, which is a $W$-group.

[felixpernegger]

This is super great thanks! I think Toronto property should be easy to add with some theorems (for ordinals) and then we completed all ordinals :)

[Moniker1998]

@StevenClontz @ccaruvana could one of you review the proximal property for this space?
It's about topological games and I'd rather an expert check if the result cited in that paper is correct

[Moniker1998]

@felixpernegger do not add the embed into W-group property for this PR. I will add it in another PR with a bunch of other spaces. #1579

[Moniker1998]

@felixpernegger Actually for the proximal property, you can also delete it. I have an argument for it which I'll add in above PR.
All of these spaces are closed subspaces of $\Sigma$-product of real lines as far as I can see and we can exploit that because pseudometrizable spaces are proximal, closed subspaces of proximal spaces are proximal and $\Sigma$-products of proximal spaces are also proximal.

[felixpernegger]

This ought to be ready now.

[prabau]

It seems most certainly fine, but can you give me the argument why a Sigma product of spaces, each of them embeddable in a W-space, is also embeddable in a W-space.

[prabau]

Suggested change

	For $a \in X$ consider the open neighborhood $U_a := [0,a+1)\subseteq X$. By the definition of $X$, $U_a$ is {P57} {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable).
	For $alpha \in X$ consider the open neighborhood $U_\alpha := [0,\alpha+1)\subseteq X$. By the definition of $X$, $U_\alpha$ is {P57} {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable).

for ordinals it's more common to use Greek letters from the beginning of the alphabet

[prabau]

Suggested change

	Consider the open cover $\mathcal{O}=\{[0,a)\mid a \in X\}$. If $X$ were weakly Lindelöf, we would find a countable $\mathcal{U}\subseteq \mathcal{O}$ such that $\bigcup \mathcal{U}$ dense. But then $\bigcup\mathcal{U}=X$, since otherwise we would find an $a \in X$ with $[a,\omega_1)\subseteq X \setminus \bigcup\mathcal{U}$. Therefore $X$ would be a countable union of countable sets and thus countable.
	Consider the open cover $\mathcal{O}=\{[0,\alpha)\mid \alpha \in X\}$.
	Every countable subset of $\omega_1$ has an upper bound in $\omega_1$.
	So if $\mathcal U$ is a countable subcollection of $\mathcal O$, there is some $\alpha\in\omega_1$ such that
	$\bigcup\mathcal U\subseteq[0,\alpha]$, which is not dense in $X$.

more direct

[Moniker1998]

@prabau look at meta-properties for W-spaces. I can't give you one, but it already exists on pi-base that W-spaces are closed under $\Sigma$-products

[prabau]

I know W-spaces are preserved by $\Sigma$-products. I was asking @felixpernegger to show me why it is also the case for "embeds into a W-group".

But there is a subtlety. Namely, if the embedding of some of the component $X_i$ maps into a W-group $G_i$ in such a way that the image does not contain the unit element of the group, there may be difficulties. Because a $\Sigma$-product is with respect to a choice of "base point" for each space, and for a Sigma product of topological groups, one has to choose the unit element as base point. If the chosen base point in each $X_i$ does not map to the unit element of $G_i$, the image of the elements in the Sigma product of the $X_i$ will not be in the Sigma product of the $G_i$ (assuming an uncountably family).

But because a topological group is homogeneous, one can compose with homeomorphisms and make the base point of $X_i$ map to the unit of $G_i$ for each $i$. And then everything works.

Please confirm.

[Moniker1998]

@prabau so you're assuming we have to choose unit element for base point for topological groups but note that on pi-base we talk about $\Sigma$-products of topological spaces.

Your objection is a valid objection but it's concerning topological groups or more precisely 'groupable spaces'.

Then the question is why $\Sigma$-product of groupable spaces is groupable. And you are right that we can use that a groupable space is homogenous, construct homeomorphism pointwise and then from definition of $\Sigma$-product it will map one into the other.

[Moniker1998]

@prabau and you might have not realized it but in above comment you wrote embeddable into W-space and not W-group

[prabau]

Ah, sorry. In my mind I meant to say W-group and wrote W-space.

Thanks for double checking.