Cocountability

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. 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 Y is cofinite.^[1]

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.^[2]

Topology

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

References

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: 33, doi:10.1007/978-1-4471-3994-2, ISBN 9781447139942