Metrizable Topology

Definition

A Topological Space $(X,\mathcal{T})$ is metrizable If there exist a Metric $d$ on $X$ that induces the topology $\mathcal{T}$.

Facts

Consider a Compact Metric Space $(X,{\mathcal{T}}_X)$ and a Hausdorff Space $(Y,{\mathcal{T}}_Y)$. If a function $f:X\to Y$ is continuous, then $Y$ is a metrizable space.