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Fréchet filter

From Encyclopedia of Mathematics
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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