The cocountable topology, also known as the countable complement topology, is a
topology that can be defined on any
infinite set . In this topology, a set is
open if its
complement in
is either countable or equal to the entire set. Equivalently, the open sets consist 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 subsets of
whose complements are countable, a property known as
cocountability. The only
closed set
In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
s in this topology are
itself and the countable subsets of
.
Definitions
Let
be an
infinite set and let
be the set of
subsets of
such that
then
is the countable complement toplogy on
, and the
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is a countable complement space.
Symbolically, the topology is typically written as
Double pointed cocountable topology
Let
be an
uncountable set
In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger t ...
. We define the topology
as all open sets whose complements are countable, along with
and
itself.
Cocountable extension topology
Let
be the real line. Now let
be the
Euclidean topology and
be the cocountable topology on
. The ''cocountable extension topology'' is the smallest topology generated by
.
Proof that cocountable topology is a topology
By definition, 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 ...
is an element of
. Similarly, the entire set
, since the
complement of
relative to itself is the empty set, which is
vacuously countable.
Suppose
. Let
. Then
by
De Morgan's laws. Since
, it follows that
and
are both countable. Because the countable union of countable sets is countable,
is also countable. Therefore,
, as its complement is countable.
Now let
. Then
again by De Morgan's laws. For each
,
is countable. The countable intersection of countable sets is also countable (assuming
is countable), so
is countable. Thus,
.
Since all three
open set axioms are met,
is a topology on
.
Properties
Every set
with the cocountable topology is
Lindelöf, since every nonempty
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
omits only countably many points of
. It is also
T1, as all singletons are closed.
If
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
are finite subsets, all compact subsets are closed, another condition usually related to the 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 and
pseudocompact, but neither
weakly countably compact nor
countably metacompact, hence not compact.
Examples
*Uncountable set: On any uncountable set, such as the real numbers
, the cocountable topology is a proper subset of the
standard topology. In this case, the topology is T
1 but not Hausdorff, first-countable, nor
metrizable.
*Countable set: If
is countable, then every subset of
has a countable complement. In this case, the cocountable topology is just the
discrete topology.
*Finite sets: On a finite set, the cocountable topology reduces to the
indiscrete topology, consisting only of the empty set and the whole set. This is because any proper subset of a finite set has a finite (and hence not countable) complement, violating the openness condition.
*Subspace topology: If
and
carries the cocountable topology, then
inherits the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
. This topology on
consists of the empty set, all of
, and all subsets
such that
is countable.
See also
*
Cofinite topology
*
List of topologies
References
{{Reflist
General topology
Topological spaces