real is second category set.

Hi:

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