# Definition:Filter Basis

## Definition

Let $S$ be a set.

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

### Definition 1

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 $\FF$.

### Definition 2

Let $\BB$ be a subset of a filter $\FF$ on $S$ such that $\BB \ne \O$.

Then $\BB$ is a filter basis of $\FF$ if and only if:

$\forall U \in \FF: \exists V \in \BB: V \subseteq U$

## Generated Filter

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

## Also known as

A filter basis is also known as a filter base.

## Linguistic Note

The plural of basis is bases.

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