The purpose of this paper is to
introduce a locally convex topology 𝒯c on a preordered topological space (X,𝒯 ) in
such a way that, if 𝒯c is weaker than 𝒯 , then it is the l.u.b. of all locally convex
topologies weaker than 𝒯 . Some of the consequences of having such a topology
defined are examined, and the concepts of c-continuity and c-limit of a function are
introduced. As an application of the machinery developed, a theorem concerning the
unique extendability of functions from dense subsets of preordered spaces into
regularly preordered spaces is established.
