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 $P \cap Q = \emptyset$?

As an alternative approach, prove (or reference the fact) that for any two distinct real numbers $x,y$, there is a rational number $q$ lying between them. If we assume (without loss of generality) that $x<y$, then $(x-1,q)$ and $(q,y+1)$ do the trick.
