Cocountable topology

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[edit intro]

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[edit]

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[edit]

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