Cofinite topology

From Citizendium, the Citizens' Compendium

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

Properties

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

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

References

• Lynn Arthur Steen; J. Arthur Seebach jr (1978). Counterexamples in Topology. Berlin, New York: Springer-Verlag, 49-50. ISBN 0-387-90312-7.