Topologies Corresponding to Continuous Representability of Preorders

A topology τ on a fixed nonempty set X is said to satisfy the weakly continuous representation property if every weakly continuous not necessarily total preorder on the topological space (X, τ) admits a continuous order preserving function. Such a property generalizes the well known continuous representation property of a topology τ on a set X (according to which every continuous total preorder on (X, τ) admits a continuous order preserving function). In this paper I present some results concerning the topologies which satisfy the weakly continuous representation property.