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 $x$?

  • 1
    $\begingroup$ isn't this answered here in a comment? $\endgroup$ – Carlo Beenakker Mar 17 '17 at 14:07
  • $\begingroup$ As far as I could notice, the comment mentions that the strictly Fréchet condition is stronger than the strongly Fréchet condition. Also, I could not find any counterexample in a first look on the linked paper, but I will read it more carefully later. Once again, thank you. $\endgroup$ – Renan Maneli Mezabarba Mar 17 '17 at 14:59

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.

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