Filter

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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.

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 $X$ . Thsi is called the cofinite filter, or Fréchet filter.

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