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>.  Thsi is called the cofinite filter, or Fréchet filter.
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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 $X$ is a structure of subsets of $X$.

Definition

Let $\mathcal{F}$ be a set of subsets of $X$. We say that $\mathcal{F}$ is a filter on $X$ if and only if each of the following conditions hold:

  • The empty set is not an element of $\mathcal{F}$
  • If $A$ and $B$ are subsets of $X$, $A$ is a subset of $B$, and $A$ is an element of $\mathcal{F}$, then $B$ is an element of $\mathcal{F}$.
  • The intersection of two elements of $\mathcal{F}$ is an element of $\mathcal{F}$.

It follows from the definition that the intersection of any finite family of elements of $\mathcal{F}$ is also an element of $\mathcal{F}$. Also, if $A$ is an element of $\mathcal{F}$, then its complement is not.

Examples

Let $Y$ be a subset of $X$. Then the set of subsets of $X$ containing $Y$ constitute a filter on $X$.

If $X$ is an infinite set, then the subsets of $X$ with finite complements constitute a filter on $X$. This is called the cofinite filter, or Fréchet filter.

See also

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