# Closure of a disjoint union of open intervals

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.

• 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? --- Any point of the real line is determined by the rational numbers (in the sense you are using, namely every real number is a limit of rational numbers), but the real line is not countable. Commented Oct 18, 2023 at 7:45