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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 ...
X. 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 X 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 X 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 X itself and the countable subsets of X.


Definitions

Let X 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 \mathcal 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 X such that H \in \mathcal \iff X \setminus H \mbox\, H = \varnothing then \mathcal is the countable complement toplogy on X , 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 ...
T = ( X , \mathcal ) is a countable complement space. Symbolically, the topology is typically written as \mathcal = \.


Double pointed cocountable topology

Let X 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 \mathcal as all open sets whose complements are countable, along with \varnothing and X itself.


Cocountable extension topology

Let X be the real line. Now let \mathcal_1 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 \mathcal_2 be the cocountable topology on X. The ''cocountable extension topology'' is the smallest topology generated by \mathcal_1 \cup \mathcal_2.


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 ...
\varnothing is an element of \mathcal. Similarly, the entire set X \in \mathcal , 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 X 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 A, B \in \mathcal. Let H = A \cap B. Then X \setminus H = X \setminus (A \cap B) = (X \setminus A) \cup (X \setminus B) by De Morgan's laws. Since A, B \in \mathcal, it follows that X \setminus A and X \setminus B are both countable. Because the countable union of countable sets is countable, X \setminus H is also countable. Therefore, H = A \cap B \in \mathcal, as its complement is countable. Now let \mathcal \subseteq \mathcal. Then X \setminus \left( \bigcup \mathcal \right) = \bigcap_ (X \setminus U) again by De Morgan's laws. For each U \in \mathcal, X \setminus U is countable. The countable intersection of countable sets is also countable (assuming \mathcal is countable), so S \setminus \left( \bigcup \mathcal \right) is countable. Thus, \bigcup \mathcal \in \mathcal. Since all three open set axioms are met, \mathcal is a topology on X.


Properties

Every set X 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 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 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 \mathbb, 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 T1 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 X is countable, then every subset of X 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 Y \subseteq X and X carries the cocountable topology, then Y 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 Y consists of the empty set, all of Y, and all subsets U \subseteq Y such that Y \setminus U 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