Complementation in the Lattice of Locally Convex Topologies

Tom Richmond* ([email protected]). Complementation in the Lattice of Locally Convex Topologies. We find all locally convex homogeneous topologies on (R,≤) and determine which of these have locally convex complements. Among the locally convex topologies on a n-point totally ordered set, each has a locally convex complement, and at least n of them have 2n−1 locally convex complements. For any infinite cardinal κ, totally ordered spaces of cardinality κ which have exactly 1, exactly κ, and exactly 2 locally convex complements are exhibited. (Received January 18, 2011)