Difference between revisions of "Filter"
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Let <math>\mathcal{F}</math> be a set of subsets of <math>X</math>. We say that <math>\mathcal{F}</math> is a filter on <math>X</math> if and only if each of the following conditions hold: | Let <math>\mathcal{F}</math> be a set of subsets of <math>X</math>. We say that <math>\mathcal{F}</math> is a filter on <math>X</math> if and only if each of the following conditions hold: | ||
| − | * The [[empty set]] is not an element of <math>\mathcal{F}</math> | + | * The [[empty set]] is not an element of <math>\mathcal{F}</math>. |
* If <math>A</math> and <math>B</math> are subsets of <math>X</math>, <math>A</math> is a subset of <math>B</math>, and <math>A</math> is an element of <math>\mathcal{F}</math>, then <math>B</math> is an element of <math>\mathcal{F}</math>. | * If <math>A</math> and <math>B</math> are subsets of <math>X</math>, <math>A</math> is a subset of <math>B</math>, and <math>A</math> is an element of <math>\mathcal{F}</math>, then <math>B</math> is an element of <math>\mathcal{F}</math>. | ||
* The intersection of two elements of <math>\mathcal{F}</math> is an element of <math>\mathcal{F}</math>. | * The intersection of two elements of <math>\mathcal{F}</math> is an element of <math>\mathcal{F}</math>. | ||
It follows from the definition that the intersection of any finite family of elements of <math>\mathcal{F}</math> is also an element of <math>\mathcal{F}</math>. Also, if <math>A</math> is an element of <math>\mathcal{F}</math>, then its [[complement]] is not. | It follows from the definition that the intersection of any finite family of elements of <math>\mathcal{F}</math> is also an element of <math>\mathcal{F}</math>. Also, if <math>A</math> is an element of <math>\mathcal{F}</math>, then its [[complement]] is not. | ||
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| + | More generally, one can define a filter on any [[Partially ordered set]] (poset) <math>(P,\leq)</math>: Let <math>F</math> be a subset of <math>P</math>. We say <math>F</math> is a filter if and only if | ||
| + | * <math>F\neq\emptyset</math>. | ||
| + | * For all <math>x,y\in F</math>, there exists <math>z\in F</math> such that <math>z\leq x</math> and <math>z\leq y</math>. | ||
| + | * If <math>x\in F</math> and <math>x\leq y\in P</math>, then <math>y\in F</math>. | ||
| + | A filter on a set <math>S</math> is a filter on the poset <math>(\mathcal{P}(S),\subseteq)</math>. | ||
== Examples == | == Examples == | ||
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* [[Ultrafilter]] | * [[Ultrafilter]] | ||
| + | [[Category:Definition]] | ||
| + | [[Category:Set theory]] | ||
{{stub}} | {{stub}} | ||
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Latest revision as of 15:59, 13 November 2024
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.
More generally, one can define a filter on any Partially ordered set (poset) 
: Let 
be a subset of 
. We say is a filter if and only if
.- For all
, there exists 
such that 
and 
. - If
and 
, then 
.
.
, there exists 
such that 
and 
.
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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