Uniform Space whose Topology is Metrizable is not necessarily Metrizable

Theorem

Let $\UU$ be a uniformity on a set $S$.

Let $\struct {\struct {S, \UU}, \tau}$ be the uniform space generated from $\UU$.

Let $T = \struct {S, \tau}$ be the uniformizable space yielded by $\struct {\struct {S, \UU}, \tau}$.

Let $T$ be a metrizable space.

Then it is not necessarily the case that $\UU$ is itself a metrizable uniformity.

Proof

Let $T = \struct {S, \tau}$ be an uncountable discrete ordinal space.

From Uncountable Discrete Ordinal Space is Metrizable, $T$ is a metrizable space.

However, from Uncountable Discrete Ordinal Space has Unmetrizable Uniformity, there exists a uniformity $\UU$ which yields the uniformizable space $T = \struct {S, \tau}$ which is not itself a metrizable uniformity.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Metric Uniformities