Finding disjoint neighborhoods of two points in $\Bbb R$ Let $x$ and $y$ be unique real numbers. How do you prove that there exists a neighborhood $P$ of $x$ and a neighborhood $Q$ of $y$ such that $P \cap Q = \emptyset$?

Without loss of generality, assume $x < y$. then
$x\in(-\infty, (x + y)/2)$ and $y \in ((x+y)/2,\infty)$.  
