Maximal complements in the lattices of pre-orders and topologies

Two topologies τ and ρ over X are said to be complementary if τ∧ρ is the indiscrete topology and τ∨ρ the discrete topology. The lattice of topologies is complemented, i.e., every topology has a complement. We will show that every AT topology (i.e., a topology such that the intersection of arbitrary many open sets is open) over a countable set has a maximal complement in the lattice of topologies. This result answers a question of S. Watson (Topology Appl. 55 (1994) 101–125). This theorem is a corollary of an analogous result for the lattice of pre-orders. We show that every pre-order P on a countable set X admits a maximal complement in the lattice of pre-orders over X. Moreover, if every connected component of P is neither discrete nor indiscrete, then such a maximal complement has all its chains of size at most two.