Nagata-Smirnov Metrization Theorem

The Nagata-Smirnov metrization theorem provides a complete characterization of metrizable spaces, removing the second-countability assumption of the Urysohn theorem. It states that a space is metrizable if and only if it is $T_3$ and has a $\sigma$ -locally finite basis.

Statement

Theorem7.10Nagata-Smirnov Metrization Theorem

A topological space $X$ is metrizable if and only if $X$ is $T_3$ (regular and Hausdorff) and has a $\sigma$ -locally finite basis, i.e., a basis $\mathcal{B} = \bigcup_{n=1}^{\infty} \mathcal{B}_n$ where each $\mathcal{B}_n$ is a locally finite family of open sets.

Motivation and Context

RemarkComparison with Urysohn

The Urysohn metrization theorem states: regular + second-countable $\implies$ metrizable. Every countable family is $\sigma$ -locally finite (it is a single locally finite family if it is locally finite, or a countable union of singletons). So the Nagata-Smirnov theorem is strictly more general.

The key insight is that second-countability (a countable basis) is replaced by the weaker condition of a $\sigma$ -locally finite basis, which allows for uncountably many basis elements organized into countably many locally finite layers.

Proof Outline

Proof

( $\Rightarrow$ : Metrizable implies $T_3$ with $\sigma$ -locally finite basis.)

Every metrizable space is $T_3$ (Theorem 7.1). For the $\sigma$ -locally finite basis: for each $n \geq 1$ , the collection of open balls $\{B(x, 1/n) : x \in X\}$ covers $X$ but may not be locally finite. However, using paracompactness (every metrizable space is paracompact), we can refine to a locally finite open cover $\mathcal{V}_n$ with each $V \in \mathcal{V}_n$ having diameter at most $2/n$ . Then $\mathcal{B} = \bigcup_n \mathcal{V}_n$ is a $\sigma$ -locally finite basis.

( $\Leftarrow$ : $T_3$ with $\sigma$ -locally finite basis implies metrizable.)

This is the deeper direction. The proof proceeds in several steps:

Step 1: $X$ is normal. (Regular + paracompact implies normal, and the existence of a $\sigma$ -locally finite basis implies paracompactness.)

Step 2: For each $B \in \mathcal{B}_n$ , use the Urysohn lemma to construct a continuous function $f_B: X \to [0, 1]$ with $f_B|_B > 0$ and $f_B|_{X \setminus B} = 0$ (well, actually $f_B|_{X \setminus U} = 0$ for a suitable $U$ ).

Step 3: Use the family of functions from Step 2 to embed $X$ into a metrizable space. The functions $\{f_B : B \in \mathcal{B}\}$ separate points and closed sets. Define an embedding $F: X \to \prod_{B \in \mathcal{B}} [0, 1]$ by $F(x) = (f_B(x))_B$ . With a carefully chosen metric on the codomain that respects the $\sigma$ -locally finite structure, this becomes an embedding into a metric space.

Step 4: Verify that $F$ is a topological embedding (injective, continuous, open onto its image).

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Applications

ExampleSpaces Metrized by Nagata-Smirnov

Discrete spaces of any cardinality: A discrete space $X$ has basis $\{\{x\} : x \in X\}$ . This is a single locally finite family (each point has a neighborhood $\{x\}$ meeting only one basis element). So discrete spaces are metrizable (with the discrete metric $d(x, y) = 1$ for $x \neq y$ ).
$\ell^2(I)$ for uncountable $I$ : The Hilbert space with uncountable index set is metrizable (normed spaces are always metrizable) and has a $\sigma$ -locally finite basis, but is not second-countable.
Disjoint unions: A disjoint union of metrizable spaces is metrizable (when the union topology is regular and has a $\sigma$ -locally finite basis).

Related Metrization Theorems

Theorem7.11Bing Metrization Theorem

A topological space is metrizable if and only if it is $T_3$ and has a $\sigma$ -discrete basis (i.e., a basis that is a countable union of discrete families, where a family is discrete if every point has a neighborhood meeting at most one member).

RemarkComparison of Metrization Theorems

| Theorem | Condition | Strength | |---|---|---| | Urysohn | $T_3$ + second-countable | Sufficient, not necessary | | Nagata-Smirnov | $T_3$ + $\sigma$ -locally finite basis | Necessary and sufficient | | Bing | $T_3$ + $\sigma$ -discrete basis | Necessary and sufficient |

The Nagata-Smirnov and Bing theorems give equivalent necessary and sufficient conditions. The Urysohn theorem is a corollary (every countable basis is $\sigma$ -locally finite).

Theorem7.12Smirnov Metrization Theorem

A topological space is metrizable if and only if it is paracompact Hausdorff and locally metrizable.

RemarkWhen Paracompactness Suffices

Smirnov's theorem adds the condition of local metrizability: every point has a metrizable neighborhood. Combined with paracompactness and Hausdorff, this gives global metrizability. This is particularly useful for manifolds: a second-countable Hausdorff manifold is paracompact and locally metrizable (locally homeomorphic to $\mathbb{R}^n$ ), hence metrizable.