Abstract
We find all locally convex homogeneous topologies on (ℝ, ≤ ) 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 ≥ 3, at least n − 2 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 2κ locally convex complements are exhibited.
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Richmond, T.A. Complementation in the Lattice of Locally Convex Topologies. Order 30, 487–496 (2013). https://doi.org/10.1007/s11083-012-9257-1
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DOI: https://doi.org/10.1007/s11083-012-9257-1