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.
$\blacksquare$