Cocountable

Cocountable is a term used in set theory and topology to describe a subset whose complement is countable. More precisely, in a space X, a subset A is cocountable if X \ A is a countable set. When X carries a topology, a subset is cocountable if its complement in X is countable, regardless of the surrounding topological structure.

A common context is the co-countable topology on an uncountable set X. In this topology, the open sets are exactly the empty set and those whose complements are countable (and X itself is open by convention). This yields a topology that is T1 (singletons are closed) but not Hausdorff: any two nonempty open sets intersect, because the union of two countable complements is countable, so their intersection is co-countable and nonempty.

Key properties include that closed sets in the co-countable topology are precisely the countable sets and X. Consequently, every uncountable subset has closure equal to X, making nonempty open sets dense. The space is compact, not first-countable, and not second-countable when X is uncountable. These features make the co-countable topology a standard example in topology of a space that is T1 but far from Hausdorff and from the usual separation axioms encountered in metric spaces.

Examples of cocountable subsets include R \ C for any countable C ⊆ R, i.e., the real line with a countable set removed. The concept also generalizes to other sets: on any uncountable X, a subset is cocountable precisely when its complement is countable.

In summary, cocountable describes a subset whose complement is countable, and in topology it often arises in the co-countable topology, illustrating a T1, compact but non-Hausdorff space.