Metrizable Spaces

A central question in point-set topology is: when can a topology be induced by a metric? Spaces whose topology arises from some metric are called metrizable. Metrization theorems provide precise conditions under which this is possible.

Definition

Definition7.1Metrizable Space

A topological space $(X, \tau)$ is metrizable if there exists a metric $d: X \times X \to [0, \infty)$ such that $\tau$ is the metric topology $\tau_d$ . That is, $U \in \tau$ if and only if for every $x \in U$ , there exists $\epsilon > 0$ with $B_d(x, \epsilon) \subseteq U$ .

We say $d$ metrizes the topology $\tau$ .

RemarkMetrizability is a Topological Property

If $X \cong Y$ and $X$ is metrizable, then $Y$ is metrizable. However, the metric itself is not a topological invariant: different metrics can induce the same topology. For example, on $\mathbb{R}^n$ , the metrics $d_1$ , $d_2$ , and $d_\infty$ all induce the same topology.

Properties of Metrizable Spaces

Theorem7.1Metrizable Spaces are Well-Behaved

Every metrizable space satisfies the following:

Hausdorff ( $T_2$ ).
Regular ( $T_3$ ).
Normal ( $T_4$ ).
Completely regular ( $T_{3\frac{1}{2}}$ ).
First-countable.
Paracompact.

Proof

(1): For $x \neq y$ , set $\epsilon = d(x, y)/2 > 0$ . Then $B(x, \epsilon) \cap B(y, \epsilon) = \emptyset$ .

(3): Let $C, D$ be disjoint closed sets. Define $f(x) = d(x, C)/(d(x, C) + d(x, D))$ . Since $C \cap D = \emptyset$ and both are closed, $d(x, C) + d(x, D) > 0$ for all $x$ . The function $f$ is continuous, $f|_C = 0$ , $f|_D = 1$ . Then $U = f^{-1}([0, 1/2))$ and $V = f^{-1}((1/2, 1])$ are disjoint open sets separating $C$ and $D$ .

(5): The collection $\{B(x, 1/n) : n \geq 1\}$ is a countable neighborhood base at each $x$ .

■

Examples

ExampleMetrizable and Non-Metrizable Spaces

Metrizable:

$\mathbb{R}^n$ with the Euclidean metric.
$\ell^p$ and $L^p$ spaces with their norm metrics.
Any countable product of metrizable spaces (with a suitable product metric).
The Hilbert cube $[0,1]^{\mathbb{N}}$ (with the metric $d(\mathbf{x}, \mathbf{y}) = \sum 2^{-n} |x_n - y_n|$ ).

Not metrizable:

The cofinite topology on an uncountable set (not Hausdorff).
The Sorgenfrey line $\mathbb{R}_\ell$ (separable but not second-countable; metrizable separable spaces are second-countable).
The long line (not second-countable).
An uncountable product $\mathbb{R}^I$ for uncountable $I$ (not first-countable).
The cocountable topology on $\mathbb{R}$ .

Obstructions to Metrizability

Theorem7.2Necessary Conditions for Metrizability

If $X$ is metrizable, then:

$X$ is Hausdorff.
$X$ is first-countable.
$X$ is paracompact.
If $X$ is separable, then $X$ is second-countable.
If $X$ is compact, then $X$ is second-countable if and only if $X$ has a countable dense subset.

ExampleThe Sorgenfrey Line is Not Metrizable

The Sorgenfrey line $\mathbb{R}_\ell$ is separable (the rationals are dense) and first-countable ( $\{[x, x + 1/n)\}$ is a countable base at $x$ ), but not second-countable.

To see this: for each $x \in \mathbb{R}$ , the set $[x, x+1)$ is open in $\mathbb{R}_\ell$ . Any basis must contain a basis element $B_x$ with $x \in B_x \subseteq [x, x+1)$ , which forces $\inf B_x = x$ . Since distinct $x$ give distinct $B_x$ , any basis has cardinality at least $|\mathbb{R}|$ , which is uncountable.

Since $\mathbb{R}_\ell$ is separable but not second-countable, it cannot be metrizable (a metrizable separable space is always second-countable).

Equivalent Metrics

Definition7.2Equivalent Metrics

Two metrics $d_1$ and $d_2$ on a set $X$ are topologically equivalent if they induce the same topology: $\tau_{d_1} = \tau_{d_2}$ .

They are uniformly equivalent if the identity maps $(X, d_1) \to (X, d_2)$ and $(X, d_2) \to (X, d_1)$ are both uniformly continuous.

They are Lipschitz equivalent if there exist constants $C_1, C_2 > 0$ with $C_1 d_1 \leq d_2 \leq C_2 d_1$ .

ExampleMetrics on $\mathbb{R}^n$

On $\mathbb{R}^n$ , the metrics $d_1(\mathbf{x}, \mathbf{y}) = \sum |x_i - y_i|$ , $d_2(\mathbf{x}, \mathbf{y}) = \sqrt{\sum (x_i - y_i)^2}$ , and $d_\infty(\mathbf{x}, \mathbf{y}) = \max |x_i - y_i|$ are all Lipschitz equivalent: $d_\infty \leq d_2 \leq d_1 \leq n \cdot d_\infty.$ Hence they induce the same topology.

RemarkStandard Bounded Metric

Every metric $d$ is topologically equivalent to the bounded metric $\bar{d}(x, y) = \min(d(x, y), 1)$ . This is useful for constructing product metrics on countable products.