Definition:Neighborhood Filter
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Definition
Let $T = \left({S, \tau}\right)$ be a topological space.
Set
Let $A \subseteq S$ such that $A \ne \O$.
Let $\NN_A$ be the set of all neighborhoods of $A$.
Then $\NN_A$ is the neighborhood filter of $A$.
Point
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 $T$).