Filter

A filter on a set $X$ is a structure of subsets of $X$ .

Definition

Let $\mathcal{F}$ be a set of subsets of $X$ . We say that $\mathcal{F}$ is a filter on $X$ if and only if each of the following conditions hold:

The empty set is not an element of $\mathcal{F}$ .
If $A$ and $B$ are subsets of $X$ , $A$ is a subset of $B$ , and $A$ is an element of $\mathcal{F}$ , then $B$ is an element of $\mathcal{F}$ .
The intersection of two elements of $\mathcal{F}$ is an element of $\mathcal{F}$ .

It follows from the definition that the intersection of any finite family of elements of $\mathcal{F}$ is also an element of $\mathcal{F}$ . Also, if $A$ is an element of $\mathcal{F}$ , then its complement is not.

More generally, one can define a filter on any partially-ordered set $(P,\leq)$ : Let $F$ be a subset of $P$ . We say $F$ is a filter if and only if

$F\neq\emptyset$ .
For all $x,y\in F$ , there exists $z\in F$ such that $z\leq x$ and $z\leq y$ .
If $x\in F$ and $x\leq y\in P$ , then $y\in F$ .

A filter on a set $S$ is a filter on the poset $(\mathcal{P}(S),\subseteq)$ .

Examples

Let $Y$ be a subset of $X$ . Then the set of subsets of $X$ containing $Y$ constitute a filter on $X$ .

If $X$ is an infinite set, then the subsets of $X$ with finite complements constitute a filter on $X$ . This is called the cofinite filter, or Fréchet filter.

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Filter

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