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 filter on $S$ (or filter of $S$) is a set $\mathcal F \subset \powerset S$ which satisfies the following conditions:

\((1)\)   $:$   \(\displaystyle S \in \mathcal F \)             
\((2)\)   $:$   \(\displaystyle \O \notin \mathcal F \)             
\((3)\)   $:$   \(\displaystyle U, V \in \mathcal F \implies U \cap V \in \mathcal F \)             
\((4)\)   $:$   \(\displaystyle \forall U \in \mathcal F: U \subseteq V \subseteq S \implies V \in \mathcal F \)             


Definition 2

A filter on $S$ (or filter of $S$) is a set $\mathcal F \subset \powerset S$ which satisfies the following conditions:

\((1)\)   $:$   \(\displaystyle S \in \mathcal F \)             
\((2)\)   $:$   \(\displaystyle \O \notin \mathcal F \)             
\((3)\)   $:$   \(\displaystyle \forall n \in \N: U_1, \ldots, U_n \in \mathcal F \implies \bigcap_{i \mathop = 1}^n U_i \in \mathcal F \)             
\((4)\)   $:$   \(\displaystyle \forall U \in \mathcal F: U \subseteq V \subseteq S \implies V \in \mathcal F \)             


Filtered Set

Let $\mathcal F$ be a filter on $S$.


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


Trivial Filter

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

If a filter $\mathcal F$ 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 $\mathcal F$, and so $\mathcal F = \powerset S$.


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


Also see

  • Results about filters can be found here.