# Definition:Filter on Set

## Definition

Let $S$ be a set.

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

### Definition 1

A set $\FF \subset \powerset S$ is a **filter on $S$** if and only if $\FF$ satisfies the filter on set axioms:

\((\text F 1)\) | $:$ | \(\ds S \in \FF \) | |||||||

\((\text F 2)\) | $:$ | \(\ds \O \notin \FF \) | |||||||

\((\text F 3)\) | $:$ | \(\ds U, V \in \FF \implies U \cap V \in \FF \) | |||||||

\((\text F 4)\) | $:$ | \(\ds \forall U \in \FF: U \subseteq V \subseteq S \implies V \in \FF \) |

### Definition 2

A set $\FF \subset \powerset S$ is a **filter on $S$** if and only if $\FF$ satisfies the filter on set axioms:

\((\text F 1)\) | $:$ | \(\ds S \in \FF \) | |||||||

\((\text F 2)\) | $:$ | \(\ds \O \notin \FF \) | |||||||

\((\text F 3)\) | $:$ | \(\ds \forall n \in \N: U_1, \ldots, U_n \in \FF \implies \bigcap_{i \mathop = 1}^n U_i \in \FF \) | |||||||

\((\text F 4)\) | $:$ | \(\ds \forall U \in \FF: U \subseteq V \subseteq S \implies V \in \FF \) |

### Filtered Set

Let $\FF$ be a filter on $S$.

Then $S$ is said to be **filtered by $\FF$**, or just a **filtered set**.

### Trivial Filter

A **filter** $\FF$ on $S$ by definition specifically does *not* include the empty set $\O$.

If a **filter** $\FF$ *were* to include $\O$, then from Empty Set is Subset of All Sets it would follow that *every* subset of $S$ would have to be in $\FF$, and so $\FF = \powerset S$.

Such a "**filter**" is called **the trivial filter** on $S$.

## Examples

### Closed Real Intervals

Let $\FF$ be the set of all closed real intervals $\closedint x y$ such that:

- $0 < x < \dfrac 1 2 < y < 1$

Then $\FF$ is a **filter** on the closed real interval $\closedint 0 1$.

## Also known as

A **filter on a set** is also known as a **filter of a set**.

## Also see

- Results about
**filters on sets**can be found**here**.