= Urysohn's metrization theorem
{c}
{title2=$T_1+\text{regular}+\text{second countable}\Longrightarrow\text{metrisable}$}
= Urysohn's metrisation theorem
{c}
{synonym}
Every <second-countable space> that is also a <T3 space> is a <metrizable space>. First <regularity of a topological space> and the <Lindelöf> property yield a <normal space>: cover each of two disjoint <closed sets> by countably many <open sets> whose closures avoid the other, and subtract the first $n$ opposite closures from the $n$th <neighborhood>. The resulting unions are disjoint <open neighborhoods>. <Urysohn's lemma> then gives, for each pair of basic sets $\overline B_n\subseteq B_m$, a <continuous function> equal to one on the smaller <closure> and zero outside the larger set. There are countably many such pairs. The shrinking form of <regularity of a topological space> shows these functions generate the topology, and $T_1$ makes them separate <points>. Their coordinate map embeds the space into the <countable> cube, with compatible <metric> $d(x,y)=\sum_j2^{-j}|f_j(x)-f_j(y)|$. The bounded tails prove compatibility with the <product topology>. Thus a <countable> base converts <separation axioms> into a numerical distance.
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