Cocountable Topology
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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. 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 and let \mathcal be the set of subsets 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 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 of X relative to itself is the empty set, which is vacuously 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. 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 \mathbb, the cocountable topology is a proper subset of the standard topology. In this case, the topology is T1 but not Hausdorff, first-countable, nor metrizable. *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. *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 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 * List of topologies


References

{{Reflist General topology Topological spaces