Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base. More explicitly, this means that 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 a union of elements of some subfamily of ${\displaystyle {\mathcal {U}}}$. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.

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