# Filter

A **filter** on a set is a structure of subsets of .

## Definition

Let be a set of subsets of . We say that is a filter on if and only if each of the following conditions hold:

- The empty set is not an element of .
- If and are subsets of , is a subset of , and is an element of , then is an element of .
- The intersection of two elements of is an element of .

It follows from the definition that the intersection of any finite family of elements of is also an element of . Also, if is an element of , then its complement is not.

More generally, one can define a filter on any Partially ordered set : Let be a subset of . We say is a filter if and only if

- .
- For all , there exists such that and .
- If and , then .

A filter on a set is a filter on the poset .

## Examples

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

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

## See also

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