
Accumulation points
Hi all,
Let us define a sequence of sets $\displaystyle (X_i)_{i \in \mathbb{Z}}$, where each set $\displaystyle X_i$ is a finite subset of the unit interval $\displaystyle [0;1]$. Let us also assume that the cardinality of $\displaystyle X_i$ grows with $\displaystyle i$.
What can you say about the set $\displaystyle \mathcal{A} \subset [0;1]$ of accumulation points of the sequence?
I call $\displaystyle A \in \mathcal{A}$ an accumulation point if any $\displaystyle \epsilon$vicinity of $\displaystyle A$ contains infinitely many points from the union $\displaystyle \bigcup_{i \in \mathbb{Z}} X_i$.
Most importantly, I would like to know if $\displaystyle \mathcal{A}$ can always be represented as a countable union of (open?/closed?) subintervals of $\displaystyle [0;1]$.
I think that another way of asking the same question would be: Is $\displaystyle \mathcal{A}$ locally compact?
To make an example: Could $\displaystyle \mathcal{A}$ be the set $\displaystyle \mathbb{Q} \cap [0;1]$ of rational numbers in $\displaystyle [0;1]$, or some set of the sort? Or is that impossible?
In either case, can you provide a proof?
Thanks,
jens