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

摘要

A necessary and suffi{}cient condition is presented for the existenceof a real continuous order-preserving function f on a topologicalpreordered space $left(X,au,preceqight)$under a resonable continuityassumption concerning the preorder $preceq$ , called extquotedblleft{}quasiICcontinuityextquotedblright{}. Under the same continuity hypotheses,a suffi{}cient condition is provided for the existence of a real continuousorder-preserving function in case that $left(X,auight)$ is acompletely regular space.