Countable complement topology

The open sets are the empty set and subsets whose complement is countable. On an uncountable set, any two nonempty open sets intersect, so the space is not Hausdorff. On a countable underlying set this topology is discrete. The compact subsets of a cocountable space are finite, regardless of the cardinality of the underlying set.

Only finite subsets are compact in the cocountable topology. An infinite subset contains distinct $c_1,c_2,\dots$. The open sets obtained by removing the tails $\{c_j:j\ge n\}$ form an increasing open cover, but every finite selection misses a tail point. Finite sets are compact in every topology.