Urysohn Metrization Theorem

Core Idea

A second-countable normal space is metrizable (its topology comes from a metric). This characterizes when abstract topological spaces are actually metric spaces. The theorem shows such spaces embed into ℓ²(ℕ), a Hilbert space with a natural metric. Provides a powerful criterion for recognizing metric spaces.

Explainer

Start with the problem. A topological space is defined abstractly by its open sets, with no requirement that distances between points exist. Metric spaces are special: they carry a distance function d(x, y) satisfying the triangle inequality, and their topology is generated by open balls. The question is: when does an abstractly defined topological space secretly have an underlying metric? Metrization means finding a metric on the space that generates exactly the given topology — not just any metric, but one that recovers all the open sets you started with.

You know two prerequisites. Normality (the T4 axiom) says that disjoint closed sets can be separated by disjoint open sets. This is a strong separation condition: any two "incompatible" closed pieces of the space can be pushed apart. Second-countability says the topology has a countable base — a countable collection of open sets from which every open set is built by unions. Euclidean spaces are second-countable (rational-radius balls centered at rational points form a countable base). The Urysohn Metrization Theorem says these two conditions together guarantee metrization.

The proof works in two steps, each using the two hypotheses in turn. First, normality gives you Urysohn functions: for any two disjoint closed sets C and D, there exists a continuous function f : X → [0,1] with f = 0 on C and f = 1 on D. Normality is exactly the condition that makes such separator functions exist. Second, second-countability gives you a countable base {U₁, U₂, U₃, …}, and for each pair (Uᵢ, Uj) with cl(Uᵢ) ⊆ Uj, normality produces a Urysohn function fᵢⱼ. Collecting all such functions gives a countable family. This family defines an embedding X → ℓ²(ℕ) by x ↦ (fᵢⱼ(x)), and the metric pulled back from ℓ² is the desired metric on X.

The theorem draws a sharp boundary between abstract topology and metric topology. All metric spaces are normal and, if second-countable, satisfy the theorem's hypotheses. So second-countable normal spaces are exactly those topological spaces that "look like" metric spaces, even when presented abstractly. Spaces that are normal but not second-countable (like the long line) may fail to be metrizable. Spaces that are second-countable but not normal (unusual but possible) also fail. The theorem is an "if and only if" in the second-countable case: for second-countable Hausdorff spaces, normality is equivalent to metrizability. This makes it one of the most satisfying classification results in topology — a clean algebraic-separation condition translating into a geometric distance structure.