Definition:Neighborhood Filter/Point

Definition
Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$.

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

Then $\mathcal N_x$ is the neighborhood filter of $x$.

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.