Cocountable topology

The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable. It follows that the only closed subsets are X and the countable subsets of X.

Every set X with the cocountable topology is Lindelöf, since every open set only omits countably many points of X.

The cocountable topology on a countable set is the discrete topology. The cocountable topology on an uncountable set is hyperconnected, thus connected, locally connected and pseudocompact, but neither weakly countably compact nor countably metacompact.

See also

• Cofinite topology

References

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995), Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, MR507446, ISBN 978-0-486-68735-3  (See example 20)