Uryshon Metrization Theorem

Definition

A second-countable Regular Space $(X,\mathcal{T})$ is metrizable.

Facts

Consider a Topological Space $(X,\mathcal{T})$. Then, the followings are equivalent:

• $X$ is second-countable Regular Space.
• $X$ might be embedded in a Hilbert Space.
• $X$ is metrizable and separable.