From Munkres,
Let X be a topological space; let $ A \subset X$. If there is a sequence of points of $A$ converging to $x$, then $x \in \overline{A}$; the converse holds if $X$ is metrizable.
Of course the converse holds if $X$ is metrizable, but I'm sure that isn't an equivalent statement. I want to know (if any) counterexamples of a non-metrizable space $X$ where the sequence lemma holds. I'm suspecting $\mathbb{R}_l$, but I have no idea on how to show the converse.