Extension of Strictly Monotonic Functions in Order-Separable Spaces

The classical Uryson-Titze theorem states that every continuous function defined on a closed subset of a normal topological space can be extended to the whole space. However, not every continuous and monotone function defined on a closed subset of a normally preordered space is extendable to the whole space. Nachbin found a necessary and sufficient condition for the existence of such an extension for nonstrictly monotone functions. This paper provides a necessary and sufficient condition for the extendability of the continuous strictly monotone functions defined on closed subsets of a normally preordered space with the separable preorder. Important examples of such spaces are the Euclidean spaces with the strict componentwise order. An application to the extension of strictly monotone preferences in Euclidean spaces is given.