Subspace of Metrizable Space is Metrizable Space

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Theorem

Let $\struct {X, \tau}$ be a metrizable topological space.

Let $\struct {Y, \tau_Y}$ be a subspace of $\struct{X, \tau}$.


Then:

$\struct {Y, \tau_Y}$ is a metrizable topological space


Proof

By definition of metrizable topological space:

there exists a metric $d$ on $X$ such that the topology induced by $d$ is $\tau$


From Metric Subspace Induces Subspace Topology:

$\tau_Y$ is the topology induced by the subspace metric $d_Y$ on $Y$

From Subspace of Metric Space is Metric Space:

$\struct {Y, d_Y}$ is a metric space


By definition, $\struct {Y, \tau_Y}$ is a metrizable topological space.

$\blacksquare$