How to show the space is Frechet? The space $X$ is monolithic and $t(X)\le \omega$, then How to show $X$ is Frechet?

$X$ is monolithic means that $\operatorname{nw}(\overline{A}) \le \kappa$ for any subset $A$ with the cardinality $\le \kappa$.

Thanks for any help.
 A: This is not too hard. Suppose $x$ is in the closure of $A$. Then there is a countable subset $B$ of $A$ such that $x$ is in the closure of $B$ (as $X$ has countable tightness). Now $\overline{B}$ has a countable network (as $X$ is ($\omega$)-monolithic). From this we should maybe get a sequence converging to $x$ from $B$...
Added This cannot be salvaged, as the counterexample by Brian shows. It does hold for compact Hausdorff monolithic spaces, as I say in the comments, because then $\overline{B}$ has a countable network and is also compact Hausdorff, and thus is metrizable. So a sequence as stated can then easily be found by first countability. Maybe Paul was working in compact spaces..
A: $\newcommand{\cl}{\operatorname{cl}}$It’s not true. Let $p$ be a free ultrafilter on $\omega$, and let $X=\omega\cup\{p\}$ as a subspace of $\beta\omega$. $X$ is countable, so it certainly has countable tightness, and $\big\{\{x\}:x\in X\big\}$ is a countable network for $X$. Finally, $p\in\cl\omega$, but no sequence in $\omega$ converges to $p$.
