In a general topological space $(X,\tau)$ we define an accumulation point $x_0$ of a set $A$ to be a point such that any open neighbourhood about $x_0$ intersects $A$.
Now it is certainly true that if a sequence $x_n\in A$ tends to some limit $x \in X$, $x$ must be an accumulation point of $A$ since $x_n$ lies in any open neighbourhood of $x$ for all $n$ sufficiently large, and so lies in the intersection of this open neighbourhood and $A$.
What I would like to know is: Are there any (preferably elementary) examples of a topological space with a subset $A$ that has an accumulation point which is not the limit of any sequence in $A$. I would also appreciate information on any conditions on a space which imply that any accumulation point of $A$ is the limit of some sequence in $A$.
For example, if $X$ is first countable (e.g. any metric space) then it is easy to show that any accumulation point of $A$ must be the limit of some sequence of points in $A$. Intuitively this is because for a given point $x$, we can find a nested sequence of open sets that "get smaller" and can eventually be contained in any open neighbourhood of $x$, so these nested open sets "contract around $x$, allowing us to find such a sequence.