Trait Suggestion: Gustin's sequence space S122 is Locally metrizable P82

[Moniker1998]

Trait Suggestion

The space Gustin's sequence space S122 is Locally metrizable P82, but this fact is not known to pi-Base today:
link to pi-Base

Proof/References

Let $U_i(\gamma)$ and $V_i(n, w)$ be as in the definition of space #125 in DOI 10.1007/978-1-4612-6290-9_6.

Since S122 is second countable, its enough to show that $U_i(\gamma)$ and $V_i(n, w)$ are regular. In fact we'll show they are zero-dimensional. For this it suffices to show that $Z = Y\cup \{(n, w)\}$ is zero-dimensional.

Suppose that $\gamma\in \text{cl}_Z V_i(n, w)$. Then $U_j(\gamma)\cap V_i(n, w)\neq\emptyset$ for all $j$. It follows that, say, $U_j(\gamma)\cap U_i(\alpha q(n, w))\neq\emptyset$ for all $j$, $\alpha\in w$. So there exists $\beta_j$ which extends both $\alpha q(n, w)$ and $\gamma$. If $\alpha q(n, w)$ extends $\gamma$, then we must have $q(n, w)\geq j$ for all $j$, which is impossible. So $\gamma$ extends $\alpha q(n, w)$ and since $\beta_j\in U_i(\alpha q(n, w))$, we must have $\gamma\in U_i(\alpha q(n, w))$. So $V_i(n, w)$ is clopen in $Z$.

Similarly, by item #2 for the space #125, we have that $\text{cl}_Z U_i(\gamma) =Z\cap \text{cl}_X(U_i(\gamma)) =Z\cap (U_i(\gamma)\cup Z(i, \gamma)) = U_i(\gamma) \cup (\{(n, w)\}\cap Z(i, \gamma))$. By taking $i$ large enough so that $i > q(n, w)$, we have $(n, w)\notin Z(i, \gamma)$ and so $U_i(\gamma)$ is clopen in $Z$.

It follows that $Z$ is regular, and so $V_i(n, w)$ and $U_i(\gamma)$ are also regular. Thus $X$ is locally metrizable.

Comment

This also resolves the property Locally pseudometrizable P144