Continuous Order-Preserving Functions on a Preordered Completely Regular Topological Space

Abstract

A necessary and suffi{}cient condition is presented for the existence of a real continuous order-preserving function f on a topological preordered space $\left(X,\tau,\preceq\right)$under a resonable continuity assumption concerning the preorder $\preceq$ , called \textquotedblleft{}quasi ICcontinuity\textquotedblright{}. Under the same continuity hypotheses, a suffi{}cient condition is provided for the existence of a real continuous order-preserving function in case that $\left(X,\tau\right)$ is a completely regular space.