Metrizable Space is not necessarily Second-Countable

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

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


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.