Difference between revisions of "Filter"
m (Link) |
(Added Categories) |
||
(One intermediate revision by one other user not shown) | |||
Line 10: | Line 10: | ||
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. | ||
− | More generally, one can define a filter on any [[Partially ordered set]] <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 | + | 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>. | * <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>. | * 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>. | ||
Line 26: | Line 26: | ||
* [[Ultrafilter]] | * [[Ultrafilter]] | ||
+ | [[Category:Definition]] | ||
+ | [[Category:Set theory]] | ||
{{stub}} | {{stub}} | ||
− | |||
− |
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 .
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 .
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
This article is a stub. Help us out by expanding it.