\[ \DeclareMathOperator{\Int}{Int} \]

A topological space is called first countable if every point has a countable neighbourhood basis .

Examples Link to heading

• A second countable space is first countable, separable and Lindelöf.
• Every metric space is first countable.
• First countability is passed on to derived topological spaces:
  • subspaces
  • product space
  • disjoint union space

Sequence Lemma Link to heading

In first countable spaces important subsets may be fully characterized by sequences.

• \(x\in \bar{A}\) if and only if \(x\) is the limit of a convergent sequence lying in \(A\).
• \(x\in \Int A\) if and only if every sequence converging to \(x\) is eventually in \(A\).
• \(A\) is closed if and only if \(A\) contains every limit of ever converging sequence of points in \(A\).
• \(A\) is open if and only if every sequence converging to a point of \(A\) is eventually in \(A\).