Definition:Neighborhood Basis

Definition
Let $$\left({X, \vartheta}\right)$$ be a topological space.

Let $$x \in X$$ be a point in $$X$$.

A neighborhood basis (or local basis) for $$x$$ is a countable set $$\mathcal V$$ of open neighborhoods of $$x$$ such that each open set of $$X$$ containing $$x$$ also contains an element of $$\mathcal V$$.