Let us define a sequence of sets , where each set is a finite subset of the unit interval . Let us also assume that the cardinality of grows with .
What can you say about the set of accumulation points of the sequence?
I call an accumulation point if any -vicinity of contains infinitely many points from the union .
Most importantly, I would like to know if can always be represented as a countable union of (open?/closed?) subintervals of .
I think that another way of asking the same question would be: Is locally compact?
To make an example: Could be the set of rational numbers in , or some set of the sort? Or is that impossible?
In either case, can you provide a proof?