Strictly Fréchet spaces versus strongly Fréchet spaces

Question

For a topological space $X$ and a point $x\in X$, consider the following definitions:

• (Gerlits and Nagy): $X$ is strictly Fréchet at $x\in X$ if for any sequence $(A_n)_{n\in\omega}$ such that $x\in\bigcap_{n\in\omega}\overline{A_n}$ there exists a sequence $(x_n)_{n\in\omega}\in\prod_{n\in\omega}A_n$ such that $x_n\to x$.
• (Siwiec): $X$ is strongly Fréchet at $x\in X$ if for any decreasing sequence $(A_n)_{n\in\omega}$ such that $x\in \bigcap_{n\in\omega}\overline{A_n}$ there exists a sequence $(x_n)_{n\in\omega}\in\prod_{n\in\omega}A_n$ such that $x_n\to x$.

Clearly, strictly Fréchet spaces are strongly Fréchet. Since these conditions are equivalent for spaces of the form $C_p(X)$, I was wondering if the converse holds in general. More precisely,

If $X$ is strongly Fréchet at $x$, then $X$ is strictly Fréchet at $x$?

Answer 1

You can find an example of a strongly Fréchet non-strictly Fréchet space in the preprint "Selective game versions of countable tightness with bounded finite selections" by L. Aurichi, A. Bella and R. Dias (see example 2.10). Basically, it's the one-point compactification of an appropriate Mrówka-Isbell $\Psi$-space.