Urysohn Metrization Theorem and Its Hypotheses

A Sufficient Test for Metrizability

A metric does more than assign distances: its open balls determine a topology. The Urysohn metrization theorem gives a condition stated entirely in topological terms that guarantees such a metric exists. Its hypotheses play different roles. Regularity lets us shrink neighborhoods without losing control of their closures; second countability lets us organize the needed separations in a countable list.

We use the convention that a space is regular if it is \(T_1\) and, whenever \(x\notin C\) with \(C\) closed, there are disjoint open sets containing \(x\) and \(C\), respectively. A \(T_1\) space is one in which every singleton is closed. A space is second countable if it has a countable basis: a countable collection \(\mathcal B\) of open sets such that for every open \(U\) and every \(x\in U\), some \(B\in\mathcal B\) satisfies \(x\in B\subseteq U\). Some texts build \(T_1\) into “regular”; others do not. Here the theorem’s \(T_1\) requirement is explicit.

The conclusion is topological: the metric need not be the metric one might initially expect, and it is not unique. The theorem asserts existence of a metric that gives exactly the original open sets, not merely a metric that separates distinct points.

What Regularity and Countability Contribute

Regularity has a useful shrinking consequence. If \(x\in U\) and \(U\) is open, apply regularity to \(x\) and the closed set \(X\setminus U\). It gives disjoint open sets \(V,W\) with \(x\in V\) and \(X\setminus U\subseteq W\). Since \(V\cap W=\varnothing\), \(V\subseteq X\setminus W\). The set \(X\setminus W\) is closed and contained in \(U\), so \(\overline V\subseteq X\setminus W\subseteq U\). Thus \(\ x\in V\subseteq\overline V\subseteq U\).

Second countability makes it possible to use only countably many such shrinkings. In the proof, one considers pairs of basis elements \(B_i,B_j\) with \(\overline{B_i}\subseteq B_j\). There are at most countably many pairs. Each suitable pair gives a continuous function that is \(0\) on \(\overline{B_i}\) and \(1\) outside \(B_j\). These functions detect the topology: if \(x\in U\), a suitable pair can be chosen with \(x\in B_i\) and \(B_j\subseteq U\). The coordinate belonging to this pair is zero at \(x\), and equals one outside \(U\).

Two standard results of normal-space theory are used in this construction. A regular second-countable space is normal, and Urysohn’s lemma says that in a normal space, disjoint closed sets can be separated by a continuous function into \([0,1]\): the function can be prescribed to be \(0\) on one closed set and \(1\) on the other. Their proofs are part of the general theory of normality and Urysohn’s lemma. Second countability implies the Lindelöf property, and the standard theorem that regular Lindelöf spaces are normal supplies the first result.

How the Functions Give a Metric

Here is the metric step of the argument. List the separating functions as \(f_1,f_2,\ldots\), with each \(f_n:X\to[0,1]\) continuous. If the list is finite, append constant zero functions. The construction above ensures both that distinct points are separated by some function and that, for each \(x\in U\) open, some coordinate is \(0\) at \(x\) and \(1\) on \(X\setminus U\). Define

The series converges because \(0\leq |f_n(x)-f_n(y)|\leq1\) and \(\sum_{n=1}^{\infty}2^{-n}=1\). The triangle inequality for absolute value, applied term by term and then summed, gives \(d(x,z)\leq d(x,y)+d(y,z)\). Symmetry and nonnegativity follow from absolute value. If \(x\ne y\), a separating coordinate has \(f_n(x)\ne f_n(y)\), so its positive contribution makes \(d(x,y)>0\). Also \(d(x,x)=0\). Thus \(d\) is a metric.

To check that its topology is the original one, first take an original open neighborhood \(U\) of \(x\). Choose a coordinate \(f_n\) with \(f_n(x)=0\) and \(f_n(y)=1\) for \(y\notin U\). For every \(y\notin U\), the \(n\)-th summand gives \(d(x,y)\geq 2^{-n}\). Hence \(B_d(x,2^{-n})\subseteq U\).

In the other direction, fix \(\varepsilon>0\). Choose \(N\) so that \(\sum_{n>N}2^{-n}<\varepsilon/2\). By continuity of the finitely many functions \(f_1,\ldots,f_N\), there is an original open neighborhood \(V\) of \(x\) on which \(|f_n(y)-f_n(x)|<\varepsilon/2\) for every \(n\leq N\). Since \(\sum_{n=1}^{N}2^{-n}<1\), for \(y\in V\), \[ d(x,y) <\frac{\varepsilon}{2}\sum_{n=1}^{N}2^{-n}+\frac{\varepsilon}{2} <\varepsilon. \] So \(V\subseteq B_d(x,\varepsilon)\). The two neighborhood comparisons show that the metric topology and original topology agree.

If \(X=\varnothing\), its unique topology is metrizable by the empty distance function; there are no pairs of points for which the metric axioms need checking. If \(X\) is nonempty, the construction above applies, including when \(X\) has just one point.

Worked Examples and the Hypotheses

What the Hypotheses Do—and Do Not—Say

Second countability gives a countable collection of coordinates, which is what allows the weighted series defining \(d\) to be used. It is a sufficient hypothesis in the theorem, not a necessary condition for every metrizable space. For example, let \(X\) be an uncountable set with the discrete topology. The distance that is \(0\) on equal pairs and \(1\) on distinct pairs induces that topology by the same ball calculation as above. Yet every basis for this topology must contain \(\{x\}\) for each \(x\): a basis element containing \(x\) and contained in the open set \(\{x\}\) can only be \(\{x\}\). Thus no countable basis exists. This example is metrizable but not second countable.

Common pitfall. A countable family of continuous functions does not automatically yield a metric. If all the coordinates on a two-point set \(X=\{a,b\}\) are constant zero, the proposed formula gives \(\ d(a,b)=\sum_{n=1}^{\infty}2^{-n}|0-0|=0\), even though \(a\ne b\). That formula is only a pseudometric in this case. The correction is to ensure the coordinates separate every pair of distinct points; the proof uses the base-pair construction to obtain this separation and also to ensure that the resulting metric topology is not finer or coarser than the original topology.