4.II.14A

Let $(M, d)$ be a metric space, and $F$ a non-empty closed subset of $M$. For $x \in M$, set

$d(x, F)=\inf _{z \in F} d(x, z)$

Prove that $d(x, F)$ is a continuous function of $x$, and that it is strictly positive for $x \notin F$.

A topological space is called normal if for any pair of disjoint closed subsets $F_{1}, F_{2}$, there exist disjoint open subsets $U_{1} \supset F_{1}, U_{2} \supset F_{2}$. By considering the function

$d\left(x, F_{1}\right)-d\left(x, F_{2}\right)$

or otherwise, deduce that any metric space is normal.

Suppose now that $X$ is a normal topological space, and that $F_{1}, F_{2}$ are disjoint closed subsets in $X$. Prove that there exist open subsets $W_{1} \supset F_{1}, W_{2} \supset F_{2}$, whose closures are disjoint. In the case when $X=\mathbf{R}^{2}$ with the standard metric topology, $F_{1}=\{(x,-1 / x): x<0\}$ and $F_{2}=\{(x, 1 / x): x>0\}$, find explicit open subsets $W_{1}, W_{2}$ with the above property.

*Typos? Please submit corrections to this page on GitHub.*