Complementation in the Lattice of Locally Convex Topologies

We find all locally convex homogeneous topologies on (a"e, a parts per thousand currency signaEuro parts per thousand) and determine which of these have locally convex complements. Among the locally convex topologies on an n-point totally ordered set, each has a locally convex complement and, for n a parts per thousand yenaEuro parts per thousand 3, at least n -aEuro parts per thousand 2 of them have 2 (n -aEuro parts per thousand 1) locally convex complements. For any infinite cardinal kappa, totally ordered spaces of cardinality kappa which have exactly 1, exactly kappa, and exactly 2 (kappa) locally convex complements are exhibited.