I have a question:

If R is the set of real numbers, and if R=\bigcup_{n}X_{n}, then there exists n such that X_{n} is dense in some open subset U of R as the set of real is second category. My question is if we have W is an open set in U. Then there are two points x and y in X_{n}\bigcap W with d(x,y)=1/2. then there is disjoint neighborhoods B(x,1/4) and B(y,1/4) about x and y 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