Cofinite topology: Difference between revisions
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In [[mathematics]], the '''cofinite topology''' is the [[topology]] on a [[set (mathematics)|set]] in the the [[open set]]s are those which have [[finite set|finite]] [[complement (set theory)|complement]], together with the empty set. Equivalently, the [[closed set]]s are the finite sets, together with the whole space. | In [[mathematics]], the '''cofinite topology''' is the [[topology]] on a [[set (mathematics)|set]] in the the [[open set]]s are those which have [[finite set|finite]] [[complement (set theory)|complement]], together with the empty set. Equivalently, the [[closed set]]s are the finite sets, together with the whole space. | ||
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==References== | ==References== | ||
* {{cite book | author=Lynn Arthur Steen | authorlink=Lynn Arthur Steen | coauthors= J. Arthur Seebach jr | title=[[Counterexamples in Topology]] | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=0-387-90312-7 }} | * {{cite book | author=Lynn Arthur Steen | authorlink=Lynn Arthur Steen | coauthors= J. Arthur Seebach jr | title=[[Counterexamples in Topology]] | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=0-387-90312-7 | pages=49-50 }} | ||
Latest revision as of 22:52, 17 February 2009
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.