# Definition:Neighborhood Filter/Point

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## Definition

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

Let $x \in S$.

Let $\NN_x$ be the set of all neighborhoods of $x$ in $T$.

Then $\NN_x$ is the neighborhood filter of $x$ (in $T$).

## Also known as

The neighborhood filter of a point can also be referred to as the system of neighborhoods or complete system of neighborhoods at that point.