Trait Suggestion: Gustin's sequence space S122 does not Have a cut point P204

[Moniker1998]

Trait Suggestion

The space Gustin's sequence space S122 does not Have a cut point P204, but this fact is not known to pi-Base today:
link to pi-Base

Proof/References

Following item #3 of space #125 in DOI 10.1007/978-1-4612-6290-9_6, if $(n, w)$ is such that $q(n, w) > \max(i, j)$, $\alpha \supset_i \gamma$ and $\beta \supset_j \gamma$, then $(n, w)\in Z(i, \gamma)\cap Z(j, \delta)$. It follows that $Z(i, \gamma)\cap Z(j, \delta)$ is infinite.

If $U_1, U_2$ are non-empty open sets of $X$, then $U_i(\gamma)\subseteq U_1$ and $U_j(\delta)\subseteq U_2$ for some $i, j$ and $\gamma, \delta\in Y$. Then by item #2 we have $\text{cl}(U_i(\gamma))\cap \text{cl}(U_j(\delta)) \supseteq Z(i, \gamma)\cap Z(j, \delta)$, and so $\text{cl}(U_1)\cap \text{cl}(U_2)$ is infinite.

If $F\subseteq X$ is finite and $U_1, U_2\subseteq X\setminus F$ are non-empty open sets, then $\text{cl}_{X\setminus F} U_1 \cap \text{cl}_{X\setminus F} U_2 = (\text{cl}_X(U_1)\cap \text{cl}_X(U_2))\setminus F$ is infinite. So $X\setminus F$ is connected. It follows that $X$ doesn't have a cut point.

Note

This resolves if S122 has properties P45 and P205