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 $S$ (or filter of $S$) 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 $S$ (or filter of $S$) 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 $S$.

Also see

• Equivalence of Definitions of Filter on Set

• Definition:Filter

• Filter on Set is Proper Filter

• Results about filters can be found here.