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


* ([Gerlits and Nagy](http://ac.els-cdn.com/0166864182900657/1-s2.0-0166864182900657-main.pdf?_tid=c2c65c2e-ff4f-11e6-b866-00000aacb35f&acdnat=1488463051_63b7512d2c12e5bb32564f78b774352a)): $X$ 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](http://ac.els-cdn.com/0016660X71901206/1-s2.0-0016660X71901206-main.pdf?_tid=6834a6e8-ff50-11e6-bd9e-00000aab0f6c&acdnat=1488463329_f2b87afb58f434d0fbee118fccc82863)): $X$ 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 $x$?
> 
> 
> 



Strictly Fréchet spaces versus strongly Fréchet spaces
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](https://arxiv.org/pdf/1603.09715.pdf)" 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.


