Let $(X,D)$ be a metric space. Suppose that $x$ and $y$ are two distinct points of $X$. Prove that there are open sets $U$ and $V$ in $X$ such that $x\in U$, $y\in V$ and $U\cap V = \emptyset$.
I know that we must show that there exists a $p_1>0$, $p_2>0$ such that for any $x,y\in X$, $x\neq y$ we have that $x\in N(x,p_1)$ and $y\in N(y,p_2)$.
Also, we must have that $x\in U$ and only in $U$ and $y\in V$ and only in $V$.