Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space ${\displaystyle T}$ is second-countable if there exists some countable collection ${\displaystyle {\mathcal {U}}=\{U_{i}\}_{i=1}^{\infty }}$ of open subsets of ${\displaystyle T}$ such that any open subset of ${\displaystyle T}$ can be written as an union of elements of some subfamily of ${\displaystyle {\mathcal {U}}}$. A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.

Many "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (Rⁿ) with its usual topology is second-countable. Although the usual base of open balls is uncountable, one can restrict to the collection of all open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis.