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Cofinite subset

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of a set $X$

A subset $A$ of $X$ for which the relative complement $X \setminus A$ is finite.

The family $\mathcal{F}(X)$ of cofinite subsets of $X$ forms a filter, known as the Fréchet filter on $X$. It is contained in any non-principal ultrafilter on $X$.

The cofinite subsets of $X$, together with the empty set, constitute the open sets of a topology on $X$, known as the cofinite topology.

How to Cite This Entry:
Cofinite subset. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cofinite_subset&oldid=39485