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