In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. In other words, Y contains all but countably many elements of X.
σ_algebras
The set of all subsets of X that are either countable or cocountable forms a σ_algebra, i.e., it is closed under the operations of countable unions, countable intersections, and complementation. This σ_algebra is the countable_cocountable algebra on X. It is the smallest σ_algebra containing every singleton set.
Topology
The cocountable topology on any set X consists of the empty set and all cocountable subsets of X. In the cocountable topology, the only closed subsets are countable sets, or the whole of X. Then X is automatically Lindelf in this topology, since every open set only omits countably many points of X.
is countable, then the countable complementtopology is just the discrete topology, as the complement of any set is countable and thus open.
Though defined similarly to the finite complementtopology, the countable complementtopology lacks many of the strong compactness properties of the finite complementtopology.
This is version 1 of countable complementtopology, born on 2004-09-24.
In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set.
The set of all subsets of X that are either countable or cocountable forms a Ï-algebra, i.e., it is closed under the operations of countable unions, countable intersections, and complementation.
The cocountabletopology (sometimes called the countable complementtopology) on any set X consists of the empty set and all cocountable subsets of X.
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