= Solution
Let $A,B$ be disjoint closed subsets of the <metric space> $(X,d)$; the cases where one is empty are immediate. The distance functions
$$
d(x,A)=\inf_{a\in A}d(x,a),
\qquad
d(x,B)=\inf_{b\in B}d(x,b)
$$
are continuous, because each is 1-Lipschitz. Their sum is strictly positive: if both distances vanished, closedness would put $x$ in $A\cap B$. Therefore
$$
u(x)=\frac{d(x,A)}{d(x,A)+d(x,B)}
$$
is continuous, equals zero on $A$, and equals one on $B$. The sets
$$
u^{-1}([0,1/3))
\quad\text{and}\quad
u^{-1}((2/3,1])
$$
are disjoint open neighbourhoods of $A$ and $B$. Thus every metric space is a <normal topological space>; this is the <normality of every metric space>.
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