Difference between revisions of "Cofinite subset"
From Encyclopedia of Mathematics
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A subset $A$ of $X$ for which the [[relative complement]] $X \setminus A$ is finite. | 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 '' | + | 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'''. | + | The cofinite subsets of $X$, together with the empty set, constitute the open sets of a [[Topological structure (topology)|topology]] on $X$, known as the '''cofinite topology'''. |
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Latest revision as of 10:14, 22 October 2016
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=39477
Cofinite subset. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cofinite_subset&oldid=39477