The number of complements of a topology on n points is at least 2n (except for some special cases) Original Research Article

We improve the results of Hartmanis (1958) and Schnare (1968,1969) by showing that, if n ⩾ 4, then any topological space on n points (equivalently, any preordered set on n points) which is not in a certain short list has at least 2n complements. We have evaluated the exact number of complements of each of the topologies in the short list.