Second-countable space

Definition

A topological space $(X, \tau)$ is second-countable when $\tau$ has a basis $\mathcal{B}$ that is countable.

Remarks

The space $\mathbb{R}^n$ is second-countable: the open balls with rational centres and rational radii form a countable basis. Each second-countable space is separable and first-countable. A metric space is second-countable if and only if it is separable. An uncountable set with the discrete topology is not second-countable.

Explanations

L1 human

A second-countable space has a countable list of open sets from which you can build every open set by unions. For $\mathbb{R}$, the open intervals with rational endpoints form such a list.