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
. 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 open set only omits countably many points of X.
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.