# Definition:Filter on Set

## Contents

## Definition

Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ denote the power set of $S$.

### Definition 1

A **filter on $S$** (or **filter of $S$**) is a set $\mathcal F \subset \mathcal P \left({S}\right)$ which satisfies the following conditions:

\((1)\) | $:$ | \(\displaystyle S \in \mathcal F \) | ||||||

\((2)\) | $:$ | \(\displaystyle \varnothing \notin \mathcal F \) | ||||||

\((3)\) | $:$ | \(\displaystyle U, V \in \mathcal F \implies U \cap V \in \mathcal F \) | ||||||

\((4)\) | $:$ | \(\displaystyle \forall U \in \mathcal F: U \subseteq V \subseteq S \implies V \in \mathcal F \) |

### Definition 2

A **filter on $S$** (or **filter of $S$**) is a set $\mathcal F \subset \mathcal P \left({S}\right)$ which satisfies the following conditions:

\((1)\) | $:$ | \(\displaystyle S \in \mathcal F \) | ||||||

\((2)\) | $:$ | \(\displaystyle \varnothing \notin \mathcal F \) | ||||||

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

\((4)\) | $:$ | \(\displaystyle \forall U \in \mathcal F: U \subseteq V \subseteq S \implies V \in \mathcal F \) |

### 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 $\varnothing$.

If a filter $\mathcal F$ *were* to include $\varnothing$, 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 $\mathcal F = \mathcal P \left({S}\right)$.

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

## Also see

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