If a set contains all its limit points must it be closed?

If a set $X$ in a topological space $T$ has the property that for all sequences $x_n \in X, x_n \to x \implies x\in X$ must X be closed? I know this is true for metric spaces but is it true for a general topological space. Here I am defining $X$ is closed iff $T\setminus X$ is open.

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$T$ being first countable should be sufficient. –  ronno Nov 12 '12 at 15:38

3 Answers

We say $x_n\to x$ iff for every open set $U\ni x$, almost all $x_n\in U$. (Note that $x_n\to x$ and $x_n\to y$ need not imply $x=y$ in general!)

Let $X=\mathbb R$ with co-countable topology, i.e. $A$ is open iff $A=\emptyset$ or $\mathbb R\setminus A$ is at most countable. Then $T:=(-\infty,0]$ contains all its limit points because for any sequence $x_n$ in $T$ and point $y>0$, the set $\mathbb R\setminus\{x_n\mid n\in\mathbb N\}$ is an open neighbourhood of $y$ not containing any members of the sequence, i.e. $y>0$ is not limit of any sequence in $T$. But $T$ is not closed because it is not countable (and not all of $\mathbb R$).

The reason is (cf. ronno's comment) that $X$ is not first countable.

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Thanks, I believe this space also has uniqueness of limits which is nice. –  Dave Nov 12 '12 at 16:08

A subset $F$ of a topological space $X$ is called sequentially closed if every limit point $x$ of some sequence $(x_n)_{n\in\mathbb N}$ in $F$ is an element of $F$. A subset $U$ of $X$ is called sequentially open if every sequence $(x_n)$ converging to a point of $U$ is eventually in $U$. It is easy to show that the complement of a sequentially open set is sequentially closed, and a bit harder to prove the other direction.

A space $X$ is called sequential if every sequentially closed subset of $X$ is closed. The other implication is always true: A closed subset is always sequentially closed. For an example of a non-sequential space see Hagen's answer.

A space with the property that for every adherence point $x$ of a subset $A$ there is a sequence $(x_n)\subseteq A$ converging to $x$ is sequential, i.e. every first-countable space is sequential.

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In general topological space you can't even talk about sequences unless you have some structure on them.

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I think we can, a sequence is just a function from the naturals to the topological space. –  Dave Nov 12 '12 at 15:31
@Dave, that's correct. A sequence is literally that, a sequence of points. Maybe he's talking about convergence? I'm not quite sure. –  Michael Dyrud Nov 12 '12 at 15:32