Extensions of continuous increasing functions

A topological preordered space is a topological space with a preorder less than or similar to, that is, a binary relation which is reflexive and transitive. Nachbin generalized Tietze's theorem and Urysohn's theorem for continuous increasing functions on a normally preordered space, where a real-valued function f is called increasing if f(x) <= f( x ') whenever x less than or similar to x '. In this paper, we provide an increasing version of Gillman-Jerison theorem characterizing C*-embedding. Namely, we show that every bounded continuous increasing function on a subspace A of a topological preordered space X can be extended over X if and only if every pair of completely order separated subsets of A can be completely order separated in X. Various theorems and several counterexamples of extensions of continuous increasing functions are also given. (C) 2023 Elsevier B.V. All rights reserved.