[Math] If every open subset of R is a disjoint union of open intervals, the number of the intervals is at most countable.

Question

Q:
Assume that every open subset of R is a disjoint union of open intervals.
Show that the number of the intervals is at most countable.

Could you give me some help to solve this problem?

Since R is uncountable, I thought that the number of the intervals is uncountable by intuition…

I think each subset's being open is a key point to prove this one, but I'm not sure how to do it though…

Answer 1

Hint : Every open interval contains a rational number. Hence if you have a collection of disjoint open intervals, then each of these intervals contains a distinct rational number. Use the fact that the set of rational numbers is countably infinite to conclude.