analysis help (countable union of disjoint sets)

Show that if is open, then can be written as an at most countable union of disjoint intervals, i.e., . (It's possible that or for some .) Hint: One way to do this is to put open intervals around each rational point in E in such a way that every point of E and only points of E are contained somewhere in these intervals. Then combine the intervals that intersect.

OK, intuitively, I get this, but what confuses me about the method they suggest is when I'm told to *combine the intervals that intersect. *Doesn't that imply that they aren't disjoint?! At firstI had the idea to take the set of all Neigborhoods of all the rationals in E of rational radius that do not intersect the complement of E, but these are not disjoint. Then I decided to take an arbitrarily large neighborhood in E and then take neighborhoods of the space that is left over and keep filling in the gaps with more and more neighborhoods until I have an at most countable amount of neighborhoods that are dense in E, but I had trouble getting that down and I also figured I should probably utilize the hint... But the hint confuses me more than the problem statement.