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]

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 $\mathcal{O}=\{[0,a)\mid a \in X\}$. If $X$ were weakly Lindelöf, we would find a countable $A\subseteq X$ such that $U = \bigcup \{[0, a) : a\in A\}$ is dense in $X$. But if $a_0 = \sup A < \omega_1$, then $U\subseteq [0, a]$, so that $U$ is contained in a proper closed set and is not dense.

[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