Second-countable space

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A topological space is second-countable if its topology admits a countable basis.

Meaning

There is one countable collection of open sets whose unions give every open set in the space. The same collection works at every point.

Examples

Euclidean space $\mathbb R^n$ has a countable basis of open balls with rational centers and positive rational radii.
A discrete space is second-countable exactly when its underlying set is countable: every singleton must occur in any basis.

Manifolds

Second countability is part of the convention used here for a topological manifold. Local Euclidean structure and Hausdorffness alone do not imply it; an uncountable disjoint union of copies of $\mathbb R$ is a counterexample.