Definition:Filter Basis/Definition 1

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Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\BB \subset \powerset S$ such that $\O \notin \BB$ and $\BB \ne \O$.

Then $\FF := \set {V \subseteq S: \exists U \in \BB: U \subseteq V}$ is a filter on $S$ if and only if:

$\forall V_1, V_2 \in \BB: \exists U \in \BB: U \subseteq V_1 \cap V_2$

Such a $\BB$ is called a filter basis of $\FF$.

Generated Filter

$\FF$ is said to be generated by $\BB$.

Also known as

A filter basis is also known as a filter base.

Also see

Linguistic Note

The plural of basis is bases.

This is properly pronounced bay-seez, not bay-siz.