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I know that if I have a sequence $ x_n $$ \to x $ where $ $$ x_n \in A $

then x is 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?

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Here is a relevant reference: – Jonas Meyer Jul 2 '11 at 22:20
More precisely, scroll down to Fréchet–Urysohn spaces... @Jonas: Could you post your comment as an answer? – François G. Dorais Jul 2 '11 at 22:57
although if there is a relationship between these spaces and what they seek, and I appreciate it because it looks very interesting, these are the places where the conditions are necessary, because here comes to the point in question ever converging is actually in the set. But at least I am not necessarily in these areas whenever we have a limit point x of set A, there exists a sequence contained in A, such that converges to x If so would appreciate the help – Daniel Jul 2 '11 at 23:12
sorry i read yet, the uryson spaces, haha thanks !!! – Daniel Jul 2 '11 at 23:24
@François: I'd rather not at the moment, but anyone who would like to include that link in an answer should feel free. – Jonas Meyer Jul 2 '11 at 23:27
up vote 6 down vote accepted

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.

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