In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a cocountable
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of a set
is a subset
whose
complement
Complement may refer to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class collections into complementary sets
* Complementary color, in the visu ...
in
is a
countable set
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 ...
. In other words,
contains all but countably many elements of
. Since the rational numbers are a countable subset of the reals, for example, the irrational numbers are a cocountable subset of the reals. If the complement is finite, then one says
is
cofinite
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 is countable, then one says the set is cocounta ...
.
σ-algebras
The set of all subsets of
that are either countable or cocountable forms a
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with a ...
, i.e., it is closed under the operations of countable unions, countable intersections, and complementation. This σ-algebra is the ''countable-cocountable algebra'' on
. It is the smallest σ-algebra containing every
singleton set
In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the a ...
.
Topology
The
cocountable topology
The cocountable topology, also known as the countable complement topology, is a topology that can be defined on any infinite set X. In this topology, a set is open if its complement in X is either countable or equal to the entire set. Equivalen ...
(also called the "countable complement topology") on any set
consists of the
empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
and all cocountable subsets of
.
References
Basic concepts in infinite set theory
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