It is known that if a space $X $ is metricable then for any subset $A $ of $X $ and a point $x \in \overline A $ there is a sequence of points in $A $ converging to $x $.
I wonder if there is an example of a topological space $X $ (not metricable then) where this doesn't always hold?
Thanks in advance!
There are many such spaces. One simple Hausdorff example is the Arens-Fort space, which is fairly clearly described in the linked article. In that space let $A$ be everything except the point $\langle 0,0\rangle$; then $\langle 0,0\rangle\in\operatorname{cl}A$, but no sequence in $A$ converges to $X$.
An even simpler example, but one that isn’t Hausdorff, is an uncountable set $X$ with the co-countable topology: the open sets are $\varnothing$ and the complements of countable sets. Let $p\in X$ be arbitrary, and let $A=X\setminus\{p\}$; then $p\in\operatorname{cl}A$, but no sequence in $A$ converges to $p$. (In fact, the only convergent sequences in $X$ are the trivial ones, i.e., the ones that are constant from some term on.)
There are also nice spaces with this property, e.g., compact Hausdorff spaces. If we let $X$ be the set of all ordinal numbers less than or equal to $\omega_1$, the first uncountable ordinal, and give $X$ the order topology, then $X$ is a compact Hausdorff space, $\omega_1\in\operatorname{cl}[0,\omega_1)$, and no sequence in $[0,\omega_1)$ converges to $\omega_1$.