The cocountable topology or countable complement topology on any set ''X'' consists of the

empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...

and all

cocountable In mathematics, a cocountable subset of a set ''X'' is a subset ''Y'' whose complement in ''X'' is a countable set. In other words, ''Y'' contains all but countably many elements of ''X''. Since the rational numbers are a countable subset of the r ...

subsets of ''X'', that is all sets whose

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...

in ''X'' is

countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...

. It follows that the only closed subsets are ''X'' and the countable subsets of ''X''. Symbolically, one writes the topology as

\mathcal = \.

Every set ''X'' with the cocountable topology is Lindelöf, since every nonempty

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...

omits only countably many points of ''X''. It is also T₁, as all singletons are closed. If ''X'' is an uncountable set then any two nonempty open sets intersect, hence the space is not Hausdorff. However, in the cocountable topology all convergent sequences are eventually constant, so limits are unique. Since compact sets in ''X'' are finite subsets, all compact subsets are closed, another condition usually related to Hausdorff separation axiom. The cocountable topology on a countable set is the discrete topology. The cocountable topology on an uncountable set is hyperconnected, thus connected,

locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness ...

and

pseudocompact In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the space be completely regular in the definition of ps ...

, but neither weakly countably compact nor countably metacompact, hence not compact.

References

* {{Citation , last1=Steen , first1=Lynn Arthur , author1-link=Lynn Arthur Steen , last2=Seebach , first2=J. Arthur Jr. , author2-link=J. Arthur Seebach, Jr. , title=

Counterexamples in Topology ''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) ha ...

, origyear=1978 , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 i ...

, location=Berlin, New York , edition=

Dover Dover () is a town and major ferry port in Kent, South East England. It faces France across the Strait of Dover, the narrowest part of the English Channel at from Cap Gris Nez in France. It lies south-east of Canterbury and east of Maidsto ...

reprint of 1978 , isbn=978-0-486-68735-3 , mr=507446 , year=1995 ''(See example 20)''. General topology

See also

References