sequence lemma, relaxing some hypothesis of a theorem I know that if I have a sequence $ x_n \to x,$
where $ x_n  \in A,$
then $x$ is a limit point of $A.$
But the converse is not always true, at least in the case of a first countable space, if so.
The question is:
Is this condition of being first countable necessary? or it´s possible to relax even more this condition?
 A: As Jonas and François pointed out, a space $X$ with the desired property (that for every $A \subseteq X$ and $x \in X$, $x \in \text{cl }A$ iff there is a sequence $\langle x_n:n \in \omega \rangle$ of points of $A$ converging to $x$) is called a Fréchet space. An important related concept is that of a a sequential space. Let $X$ be a space. A set $A \subseteq X$ is sequentially closed if whenever $\langle x_n:n \in \omega \rangle$ is a sequence of points of $A$ and $\langle x_n:n \in \omega \rangle \to x$ in $X$, then $x \in A$. $X$ is sequential if every sequentially closed subset of $X$ is closed. A set $A \subseteq X$ is sequentially closed if whenever $\langle x_n:n \in \omega \rangle$ is a sequence of points of $A$ and $\langle x_n:n \in \omega \rangle \to x$ in $X$, then $x \in A$.
The Fréchet property is strictly weaker than first countability and strictly stronger than the property of being a sequential space. In fact, a space is Fréchet iff it is hereditarily sequential. Within the class of sequential spaces, Fréchet spaces can be characterized as those that do not contain a copy of the space $Y$ described here in Dan Ma's Topology Blog. In fact Dan Ma's Topology Blog discusses these properties in considerable detail. Start here for another description of $Y$ and related discussion and follow the links to older posts on sequential spaces.
