The cocountable topology, also known as the countable complement topology, is a
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
that can be defined on any
infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set ...
. In this topology, a set is
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979
* ''Open'' (Go ...
if its
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 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
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set ...
and let
be the set of
subsets
In mathematics, a set ''A'' is a subset of a set ''B'' if all 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 are unequal, then ''A'' is a proper subse ...
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
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot ...
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
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 ...
of
relative to itself is the empty set, which is
vacuously
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied.
It is sometimes said that a s ...
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
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 to ...
. 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 subsets (whether disjoint or non-disjoint). The name ''irreducible space'' is ...
, 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 of open connected sets.
As a stronger notion, the space ''X'' is locally path connected if ev ...
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 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.
Examples
*Uncountable set: On any uncountable set, such as the real numbers
, the cocountable topology is a proper subset of the
standard topology
In mathematics, the real coordinate space or real coordinate ''n''-space, of dimension , denoted or , is the set of all ordered -tuples of real numbers, that is the set of all sequences of real numbers, also known as ''coordinate vectors''.
S ...
. In this case, the topology is T
1 but not Hausdorff, first-countable, nor
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
.
*Countable set: If
is countable, then every subset of
has a countable complement. In this case, the cocountable topology is just 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 to ...
.
*Finite sets: On a finite set, the cocountable topology reduces to the
indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
, 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
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 ...
*
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
{{Reflist
General topology
Topological spaces