Difference between revisions of "Fréchet filter"
From Encyclopedia of Mathematics
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| + | The [[filter]] on an infinite set $A$ consisting of all [[cofinite subset]]s 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 filter|principal]]. | ||
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| + | 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. | ||
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| + | ====References==== | ||
| + | <table> | ||
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> Thomas Jech, ''Set Theory'' (3rd edition), Springer (2003) {{ISBN|3-540-44085-2}} {{ZBL|1007.03002}}</TD></TR> | ||
| + | </table> | ||
Latest revision as of 16:20, 18 November 2023
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=39479
Fréchet filter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_filter&oldid=39479