# Definition:Local Basis/Neighborhood Basis of Open Sets

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## Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $x$ be an element of $S$.

A **local basis** at $x$ is a set $\mathcal B$ of open neighborhoods of $x$ such that every neighborhood of $x$ contains a set in $\mathcal B$.

That is, a **local basis** at $x$ is a neighborhood basis of $x$ consisting of open sets.