Hi:

I have a question:

If is the set of real numbers, and if , then there exists such that is dense in some open subset of as the set of real is second category. My question is if we have is an open set in . Then there are two points and in with . then there is disjoint neighborhoods and about and respectively.

I want to be sure if we could find really two points with distance 1/2.

Every guidance is highly appreciated.

Thaaaaaaank you in advance