Uniform Space whose Topology is Metrizable is not necessarily Metrizable

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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.


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.