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In mathematics, a cocountable subset of a set is a subset whose complement in is a countable set. In other words, contains all but countably many elements of . Since the rational numbers are a countable subset of the reals, for example, the irrational numbers are a cocountable subset of the reals. If the complement is finite, then one says is cofinite.[1]

σ-algebras

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The set of all subsets of 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 . It is the smallest σ-algebra containing every singleton set.[2]

Topology

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The cocountable topology (also called the "countable complement topology") on any set consists of the empty set and all cocountable subsets of .[3]

References

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  1. ^ Halmos, Paul; Givant, Steven (2009), "Chapter 5: Fields of sets", Introduction to Boolean Algebras, Undergraduate Texts in Mathematics, New York: Springer, pp. 24–30, doi:10.1007/978-0-387-68436-9_5, ISBN 9780387684369
  2. ^ Halmos & Givant (2009), "Chapter 29: Boolean σ-algebras", pp. 268–281, doi:10.1007/978-0-387-68436-9_29
  3. ^ James, Ioan Mackenzie (1999), "Topologies and Uniformities", Springer Undergraduate Mathematics Series, London: Springer, p. 33, doi:10.1007/978-1-4471-3994-2, ISBN 9781447139942{{citation}}: CS1 maint: work parameter with ISBN (link)


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Cocountable topology

of the empty set and all subsets of X {\displaystyle X} whose complements are countable, a property known as cocountability. The only closed sets in this

Filter on a set

{\displaystyle X} is a set, the cocountable subsets of X {\displaystyle X} (those whose complement is countable) form a filter, the cocountable filter which is finer

Compact space

compact since the collection of open subsets {{0, x} : x ∈ X} does not have a finite subcover. For the cocountable topology on a space with uncountably

General topology

the smallest T1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or

Cofiniteness

complement is not finite, but is countable, then one says the set is cocountable. These arise naturally when generalizing structures on finite sets to

Hausdorff space

Hausdorff is the cofinite topology defined on an infinite set, as is the cocountable topology defined on an uncountable set. Pseudometric spaces typically

Limit point compact

limit point compact. An example is given by an uncountable set with the cocountable topology. Every normal pseudocompact space is limit point compact. Proof:

List of topologies

− All subsets are open. Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are open. Cocountable topology