Topologies corresponding to continuous representability of preorders

Abstract

A topology τ on a _xed 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.