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?