A pseudocompact space with $G_\delta$-point Is evert $T_2$ pseudocompact space with $G_\delta$-points always  first countable? Does there exist a counterexample?
Thanks ahead.
 A: I recall that a space $X$ is called pseudocompact in our mathematical school or
feebly compact, for instance, in Mikhail Tkachenko's school provided
each locally finite family of nonempty open subsets of the space $X$ is
finite.
Yesterday I have posed this question at the divertissement at
our seminar “Topology & Applications”
and then my friend Oleg Gutik and me answered it. 
We construct an example as follows. Let $X_0$ be a
non-empty $T_1$ space. Determine a topology on the set
$X=(X_0\times\omega)\cup\{y_0\}$, where $y_0\not\in X_0\times\omega$ by the following base
$\mathcal B=\{U\times \{n\}: U$ is an open subset of the space $X_0, n\in\omega \}\cup
\{\{y_0\}\cup\bigcup_{m\ge n}X_0\setminus F_m: n\in\omega$, $F_m$ is a finite subset of the space $X_0$ for each $m\in\omega$ such that $m\ge n$}.
It is easy to check the following:


*

*the space $X$ is Hausdorff provided the space $X_0$ is Hausdorff;

*the space $X$ is pseudocompact provided the space $X_0$ is a pseudocompact space without isolated
points;

*each point of the space $X$ is $G_\delta$ provided each point of the space $X_0$ is $G_\delta$;

*the point $y_0$ has uncountable character provided the space $X_0$ is infinite.
So this construction yields the negative answer to you question if we take the unit segment as $X_0$.
