Metrization theorems ask when an abstract topology comes from a metric. They identify which countability and separation assumptions make topology quantitatively measurable.
Metrics and induced topology
Every metric gives a topology, but many topologies are not metric.
Definition. A metric \(d\) on \(X\) induces the topology whose basis is the family of open balls \(B_d(x,\varepsilon)=\{y:d(x,y)<\varepsilon\}\). A space is metrizable if its topology is induced by some metric.
Proposition. Every metrizable space is Hausdorff, regular, and first countable.
Proof. Distinct points have positive distance, and disjoint balls of radius less than half that distance separate them. If \(x\notin F\) with \(F\) closed, then \(d(x,F)>0\) need not hold globally, but because \(F\) is closed there is some ball around \(x\) disjoint from \(F\); smaller concentric balls separate \(x\) and \(F\). The balls \(B(x,1/n)\) form a countable local basis.
Urysohn metrization theorem
Second countability plus regularity produces enough continuous functions to embed into a metric cube.
Theorem. Every regular second-countable \(T_1\) space is metrizable.
Proof. Let \(\{B_n\}\) be a countable basis. For each pair \(B_i,\overline{B_i}\subseteq B_j\), Urysohn-type regularity in a regular second-countable space gives a continuous function \(f_{ij}:X\to[0,1]\) that is \(1\) on \(\overline{B_i}\) and \(0\) outside \(B_j\). The countable family of such functions separates points and closed sets. The map \(F:X\to[0,1]^{\mathbb N}\) collecting these functions is an embedding. Since the Hilbert cube is metrizable, the subspace \(F(X)\) is metrizable, and the pulled-back metric metrizes \(X\).
Remark. A common statement assumes normality; regular second-countable \(T_1\) spaces are normal, so the function construction is available.
Nagata-Smirnov theorem
Second countability can be replaced by a locally finite basis condition.
Theorem. A regular \(T_1\) space is metrizable if and only if it has a \(\sigma\)-locally finite basis.
Proof. One direction follows from metric spaces admitting \(\sigma\)-locally finite refinements by balls at dyadic scales. For the converse, a \(\sigma\)-locally finite basis gives countably many locally finite covers. Regularity lets one shrink these covers, and the associated distance-to-complement functions separate points from closed sets. Combining the countably many pseudometrics into \(d(x,y)=\sum_n2^{-n}\min(1,d_n(x,y))\) gives a metric inducing the original topology.
Remark. This theorem is the natural metrization result for spaces larger than second countable spaces.
Moore and paracompact variants
Metrization theory is a map of how local bases, refinements, and separation interact.
Definition. A development on \(X\) is a sequence of open covers \(\mathcal U_n\) such that for every \(x\), the stars \(\operatorname{St}(x,\mathcal U_n)\) form a local basis at \(x\).
Theorem. A regular space with a \(\sigma\)-locally finite development is metrizable.
Proof. The development gives controlled local neighborhoods, and \(\sigma\)-local finiteness permits the construction of continuous functions or pseudometrics subordinate to the covers. Summing the resulting pseudometrics gives a metric whose balls refine and are refined by the star neighborhoods, hence induces the original topology.
Example. Many function spaces are not metrizable because they fail first countability or have no countable/local-finite basis compatible with the relevant topology.