Cocountable topology
Encyclopedia
The cocountable topology or countable complement topology on any set X consists of the empty set
and all cocountable
subsets of X, that is all sets whose complement
in X is countable
. It follows that the only closed subsets are X and the countable subsets of X.
Every set X with the cocountable topology is Lindelöf
, since every nonempty open set
omits only countably many points of X. It is also T1
, as all singletons are closed. The only compact
subsets of X are the finite subsets, so X has the property that all compact subsets are closed, even though it is not Hausdorff
if uncountable.
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
.
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
and all cocountable
Cocountable
In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. In other words, Y contains all but countably many elements of X. For example, the irrational numbers are a cocountable subset of the reals...
subsets of X, that is all sets whose complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
in X is countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
. It follows that the only closed subsets are X and the countable subsets of X.
Every set X with the cocountable topology is Lindelöf
Lindelöf space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
, since every nonempty open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
omits only countably many points of X. It is also T1
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
, as all singletons are closed. The only compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
subsets of X are the finite subsets, so X has the property that all compact subsets are closed, even though it is not Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
if uncountable.
The cocountable topology on a countable set is the discrete topology. The cocountable topology on an uncountable set is hyperconnected
Hyperconnected space
In mathematics, a hyperconnected space is a topological space X that cannot be written as the union of two proper closed sets. The name irreducible space is preferred in algebraic geometry....
, thus connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
, locally connected
Locally connected space
In topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.-Background:...
and pseudocompact
Pseudocompact space
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.-Properties related to pseudocompactness:...
, but neither weakly countably compact
Limit point compact
In mathematics, a topological space X is said to be limit point compact or weakly countably compact if every infinite subset of X has a limit point in X. This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactness, and sequential compactness are...
nor countably metacompact
Metacompact space
In mathematics, in the field of general topology, a topological space is said to be metacompact if every open cover has a point finite open refinement...
.