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| Double-pointed cofinite topology ： ウィキペディア英語版 |
Cofiniteness
In mathematics, a cofinite subset of a set ''X'' is a subset ''A'' whose complement in ''X'' is a finite set. In other words, ''A'' contains all but finitely many elements of ''X''. If the complement is not finite, but it is countable, then one says the set is cocountable.
These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product topology or direct sum.
==Boolean algebras==
The set of all subsets of ''X'' that are either finite or cofinite forms a Boolean algebra, i.e., it is closed under the operations of union, intersection, and complementation. This Boolean algebra is the finite-cofinite algebra on ''X''. A Boolean algebra ''A'' has a unique non-principal ultrafilter (i.e. a maximal filter not generated by a single element of the algebra) if and only if there is an infinite set ''X'' such that ''A'' is isomorphic to the finite-cofinite algebra on ''X''. In this case, the non-principal ultrafilter is the set of all cofinite sets.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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