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 20:55, 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
.
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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