Cocountable topology

In mathematics, the cocountable topology is the topology on a set in which the open sets are those which have countable complement, together with the empty set. Equivalently, the closed sets are the countable sets, together with the whole space.

Properties

If X is countable, then the cocountable topology on X is the discrete topology, in which every set is open. We therefore assume that X is an uncountable set with the cocountable topology; it is:

• connected, indeed hyperconnected;
• T1 but not Hausdorff.

References

• Lynn Arthur Steen; J. Arthur Seebach jr (1978). Counterexamples in Topology. Berlin, New York: Springer-Verlag, 50-51. ISBN 0-387-90312-7.