# Fréchet filter

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The filter on an infinite set \$A\$ consisting of all cofinite subsets of \$A\$: that is, all subsets of \$A\$ such that the relative complement is finite. More generally, the filter on a set \$A\$ of cardinality \$\mathfrak{a}\$ consisting of all subsets of \$A\$ with relative complement of cardinality strictly less than \$\mathfrak{a}\$. The Fréchet filter is not principal.

The Fréchet ideal is the ideal dual to the Fréchet filter: it is the collection of all finite subsets of \$A\$, or all subsets of cardinality strictly less than \$\mathfrak{a}\$, respectively.

#### References

 [1] Thomas Jech, Set Theory (3rd edition), Springer (2003) ISBN 3-540-44085-2 Zbl 1007.03002
How to Cite This Entry:
Fréchet filter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_filter&oldid=39482