## Accumulation points

Hi all,

Let us define a sequence of sets $(X_i)_{i \in \mathbb{Z}}$, where each set $X_i$ is a finite subset of the unit interval $[0;1]$. Let us also assume that the cardinality of $X_i$ grows with $i$.

What can you say about the set $\mathcal{A} \subset [0;1]$ of accumulation points of the sequence?

I call $A \in \mathcal{A}$ an accumulation point if any $\epsilon$-vicinity of $A$ contains infinitely many points from the union $\bigcup_{i \in \mathbb{Z}} X_i$.

Most importantly, I would like to know if $\mathcal{A}$ can always be represented as a countable union of (open?/closed?) subintervals of $[0;1]$.
I think that another way of asking the same question would be: Is $\mathcal{A}$ locally compact?
To make an example: Could $\mathcal{A}$ be the set $\mathbb{Q} \cap [0;1]$ of rational numbers in $[0;1]$, or some set of the sort? Or is that impossible?

In either case, can you provide a proof?

Thanks,
jens