Metrizable Space is not necessarily Second-Countable

Theorem

Let $T = \struct {S, \tau}$ be a topological space which is metrizable.

Then it is not necessarily the case that $T$ is second-countable.

Proof

Let $T$ be an uncountable discrete space.

From Standard Discrete Metric induces Discrete Topology, $T$ is metrizable.

From Uncountable Discrete Space is not Second-Countable, $T$ is not second-countable.

$\blacksquare$

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: Metrizability