Cocountable topology: Difference between revisions
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In [[mathematics]], the '''cocountable topology''' is the [[topology]] on a [[set (mathematics)|set]] in which the [[open set]]s are those which have [[countable set|countable]] [[complement (set theory)|complement]], together with the empty set. Equivalently, the [[closed set]]s are the countable sets, together with the whole space. | In [[mathematics]], the '''cocountable topology''' is the [[topology]] on a [[set (mathematics)|set]] in which the [[open set]]s are those which have [[countable set|countable]] [[complement (set theory)|complement]], together with the empty set. Equivalently, the [[closed set]]s are the countable sets, together with the whole space. | ||
Revision as of 21:48, 17 February 2009
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.