# Definition:Filter Basis

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## 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**.

## Also see

## Linguistic Note

The plural of **basis** is **bases**.

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