Trait Suggestion: Gustin's sequence space S122 is not Locally connected P41

[Moniker1998]

Trait Suggestion

The space Gustin's sequence space S122 is not Locally connected P41, but this fact is not known to pi-Base today:
link to pi-Base

Proof/References

Let $U_i(\alpha)$ be defined as in space #125 in DOI 10.1007/978-1-4612-6290-9_6.

By item #2 we have $\text{cl}_X(U_i(\gamma)) = U_i(\gamma)\cup Z(i, \gamma)$. From this, $\text{cl}_Y(U_i(\gamma)) = (U_i(\gamma)\cup Z(i, \gamma))\cap Y = U_i(\gamma)$ so that $U_i(\gamma)$ is clopen in $Y$.

Suppose that $X$ is locally connected. Since $Y$ is open in $X$, if $\gamma\in Y$ then there is open connected $U$ with $\gamma\in U\subseteq Y$. There is $i$ such that $U_i(\gamma)\subseteq U$. Since $U_{i+1}(\gamma)$ is a proper subset of $U_i(\gamma)$, $U_{i+1}(\gamma)$ is a non-empty proper clopen subset of $U$. This contradicts that $U$ is connected.