real is second category set.

Hi:

I have a question:

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