# Definition:Filter on Set

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## 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.