Difference between revisions of "Filter"
(New page: A '''filter''' on a set <math>X</math> is a structure of subsets of <math>X</math>. == Definition == Let <math>\mathcal{F}</math> be a set of subsets of <math>X</math>. We say t...) |
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Let <math>Y</math> be a subset of <math>X</math>. Then the set of subsets of <math>X</math> containing <math>Y</math> constitute a filter on <math>X</math>. | Let <math>Y</math> be a subset of <math>X</math>. Then the set of subsets of <math>X</math> containing <math>Y</math> constitute a filter on <math>X</math>. | ||
| − | If <math>X</math> is an [[infinite | infinite set]], then the subsets of <math>X</math> with finite complements constitute a filter on <math>X</math>. | + | If <math>X</math> is an [[infinite | infinite set]], then the subsets of <math>X</math> with finite complements constitute a filter on <math>X</math>. This is called the cofinite filter, or Fréchet filter. |
== See also == | == See also == | ||
Revision as of 19:55, 13 October 2019
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 .
are subsets 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.
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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