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