Metrizable space

Definition

A topological space $(X, \tau)$ is metrizable when there is a metric $d$ on $X$ whose metric topology is $\tau$.

Remarks

The metric is not unique: equivalent metrics give the same topology. Each metrizable space is Hausdorff and normal. An indiscrete space with at least two points is not metrizable. Urysohn's metrization theorem: each second-countable regular Hausdorff space is metrizable.

Explanations

L1 human

A metrizable space is a topological space where you can measure distances in a way that gives exactly the open sets you already have. Properties that hold in all metric spaces then hold in this space.