Examples of $\kappa$-Fréchet-Urysohn spaces.

Question

We say that

A space $X$ is $\kappa$-Fréchet-Urysohn at a point $x\in X$, if whenever $x\in\overline{U}$, where $U$ is a regular open subset of $X$, some sequence of points of $U$ converges to $x$.

I'm looking for some examples of $\kappa$-Fréchet-Urysohn space. I guess it is not true that every compact Hausdorff space is a $\kappa$-Fréchet-Urysohn But how about compact Hausdorff homogeneous spaces?

Answer 1

It may be helpful for you:

Every first countable space $X$ is always $\kappa$-Fréchet-Urysohn spaces.

Proof: As $X$ is first countable, then $\overline{U}$ is also first countable. For any $x\in \overline{U}$, as $U$ is dense in $\overline{U}$, then there exists a sequence of $U$ such converges to $x$.

Answer 2

What examples are you looking for?

I think, for some exotic examples you should search Engelking’s “General Topology” (on Frechet-Urysohn spaces) and part 10 “Generalized metric spaces” of the “Handbook of Set-Theoretic Topology”.