Sobriety of ω-well-filtered Fréchet spaces

Determine whether every ω-well-filtered Fréchet topological space is sober; that is, ascertain if the class of ω-well-filtered spaces that are Fréchet necessarily satisfies sobriety.

Background

The paper establishes a key sufficient condition: if X is an ω-well-filtered coherent d-space and X×X is a Fréchet space, then X is sober (Theorem 3.9). This leverages Fréchet convergence to show irreducible closed sets are directed and yield generic points.

It is also noted that if X×X is a Fréchet space then X itself is Fréchet, since X is a retract of X×X (Proposition 3.14). Furthermore, first-countable well-filtered spaces are known to be sober, and first-countability implies the Fréchet property. Motivated by these, the authors pose whether the coherence and d-space hypotheses can be removed, asking if every ω-well-filtered Fréchet space is sober.