My question mainly is wether or not the closure of a disjoint union of open intervals is the union of the closure of those intervals, because I cannot think of any counterexample, and if this is the case I don't understand intuitively why this is the case, because if we consider the complement of the cantor set on $[0,1]$ this set consists of a disjoint union of open intervals and it's closure is the whole interval $[0,1]$, but how can this be? Any point of the cantor set would be determined by the intervals of which is an endpoint, but this would lead into the cantor set being countable, so what am I missing?
I also know that the interval $[0,1]$ cannot be written as a disjoint union of closed sets, but this doesn't really give an answer to my question.
Suppose you have a sequence of points, one in each of a countable number of intervals. Those points might converge to something not in any of the interval closures. As an example:
Take the interval of size 0.5 centered at 1
Then the interval of size 0.25 centered at 1/2
Then the interval of size 0.125 centered at 1/4
and so on. The limit of the center-points of those intervals (namely 0) is in the closure of the union, but it's not in the closure of any one interval.