Trait Suggestion: Ordinal spaces S35, S36 are Cozero complemented P61

[Moniker1998]

Trait Suggestion

The Ordinal spaces S35, S36 are Cozero complemented P61, but this fact is not known to pi-Base today: link to pi-Base 1 link to pi-Base 2

Proof/References

A cofinal zero-set $Z\subseteq \omega_1$ in $\omega_1$ must contain the tail $[\alpha+1, \omega_1)\subseteq Z$ as is well-known. Since compact $G_\delta$ sets are zero-sets (alternatively, closed $G_\delta$ sets in normal space are zero-sets), we see that zero-sets of $\omega_1$ are precisely those closed sets $Z$ for which there exists $\alpha < \omega_1$ with $Z\subseteq [0, \alpha]$, or $[\alpha+1, \omega_1)\subseteq Z$.

It follows that clopen sets of $\omega_1$ are either open bounded sets or those which contain $[\alpha+1, \omega_1)$ for some $\alpha < \omega_1$. The same argument shows that clopen sets of $\omega_1+1$ are either open sets bounded by some $\alpha < \omega_1$ or those which contain $[\alpha+1, \omega_1]$ for some $\alpha < \omega_1$.

Since a countable Tychonoff space has to be cozero complemented, if $U \subseteq [0, \alpha]$ for some $\alpha < \omega_1$, find open $V\subseteq [0, \alpha]$ such that $U, V$ are disjoint and $U\cup V$ is dense in $[0, \alpha]$. Then $V_0 = V\cup [\alpha+1, \omega_1)$ is a cozero set of $\omega_1$ such that $U, V_0$ are disjoint and $U\cup V_0$ is dense in $\omega_1$.

Similarly if $[\alpha+1, \omega_1)\subseteq U$ then write $U = U_0\cup [\alpha+1, \omega_1)$ where $U_0\subseteq [0, \alpha]$ is open and find open $V\subseteq [0, \alpha]$ with $U_0\cup V$ dense in $[0, \alpha]$ and $U_0, V$ disjoint, then $U, V$ are disjoint and $U\cup V$ is dense in $\omega_1$.

The exact same argument for $\omega_1+1$.