Topological Space Homeomorphic to Metrizable Space is Metrizable Space

Theorem
Let $\struct{X_1, \tau_1}$ be a metrizable topological space.

Let $\struct{X_2, \tau_2}$ be a topological space homeomorphic to $\struct{X_1, \tau_1}$.

Then:
 * $\struct{X_2, \tau_2}$ is a metrizable topological space

Proof
By definition of metrizable topological space:
 * there exists a metric $d_1$ on $X_1$ such that the topology induced by $d_1$ is $\tau_1$

Hence:
 * $\struct{X_2, \tau_2}$ is homeomorphic to $\struct{X_1, \tau_1}$ where the topology $\tau_1$ is the topology induced by the metric $d_1$

By definition, $\struct{X_2, \tau_2}$ is a metrizable topological space.