Cocountable topology on arvau

Wikipedia / Cocountable topology

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 is either countable or equal to the entire set. Equivalently, the open sets consist of the empty set whose complements are countable, a property known as cocountability. The only closed sets.

Definitions

be an infinite set be the set of subsets, and the topological space is a countable complement space.

Double pointed cocountable topology

be an uncountable set itself.

Cocountable extension topology

be the Euclidean topology. The cocountable extension topology.

Proof that cocountable topology is a topology

By definition, the empty set, since the complement relative to itself is the empty set, which is vacuously countable.

. Then

by De Morgan's laws, as its complement is countable.

. Then

Since all three open set axioms.

Properties

with the cocountable topology is Lindelöf, since every nonempty open set. It is also T1, as all singletons are closed.

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 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

, 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.
has a countable complement. In this case, the cocountable topology is just the discrete topology.
inherits the subspace topology is countable.