Trait Suggestion: Gustin's sequence space S122 is not T_2.5 P4

[Moniker1998]

Trait Suggestion

The space Gustin's sequence space S122 is not Urysohn P4, but this fact is not known to pi-Base today:
link to pi-Base

Proof/References

If $U_1, U_2$ are open non-empty sets, then $\text{cl}(U_1)\cap \text{cl}(U_2)\neq\emptyset$.

See item #3 for space #125 in DOI 10.1007/978-1-4612-6290-9_6.

Comment

This property looks like something that should be called anti-Urysohn space, but I couldn't find that property in literature or in pi-base.

[Moniker1998]

Seem like this was considered by @StevenClontz already, but perhaps forgotten?

[prabau]

I am not necessarily recommending to add this "anti-Urysohn" property, but some people seem to have called that "Brown space" if I am not mistaken.

https://zbmath.org/?q=ut%3ABrown+space+cc%3A54

However, it seems that in some of the papers, T. Banakh has called a Brown space one in which the intersection of the closures of nonempty open sets is infinite? Would need to research first in the literature exactly what was used and in what context.