Cocountability
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
[edit]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
[edit]The cocountable topology (also called the "countable complement topology") on any set consists of the empty set and all cocountable subsets of .[3]
References
[edit]- ↑ 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
- ↑ Halmos & Givant (2009), "Chapter 29: Boolean σ-algebras", pp. 268–281, doi:10.1007/978-0-387-68436-9_29
- ↑ 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
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