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 oth ...
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 ...
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-clas ...
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 number ...
. It follows that the only closed subsets are ''X'' and the countable subsets of ''X''. Symbolically, one writes the topology as
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 a ...
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
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest t ...
. The cocountable topology on an uncountable set is
hyperconnected
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...
, thus
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
,
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, connectedne ...
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 p ...
, but neither
weakly countably compact In mathematics, a topological space ''X'' is said to be limit point compact or weakly countably compact if every infinite subset of ''X'' has a limit point in ''X''. This property generalizes a property of compact spaces. In a metric space, limit ...
nor
countably metacompact
In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an op ...
, hence not compact.
See also
*
Cofinite topology
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocou ...
*
List of topologies
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that ...
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) ...
, 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 ...
, 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 Maidstone ...
reprint of 1978 , isbn=978-0-486-68735-3 , mr=507446 , year=1995 ''(See example 20)''.
General topology