Nonlinear order isomorphisms on function spaces

TL;DR

This paper characterizes order isomorphisms between various function spaces on topological and metric spaces, extending classical results to noncompact spaces and different function classes.

Contribution

It introduces near vector lattices and extends order isomorphism characterizations to noncompact spaces and diverse function spaces.

Findings

Order isomorphisms imply homeomorphisms between underlying spaces.

Characterizations for spaces of uniformly continuous, Lipschitz, and differentiable functions.

Extension of classical results to noncompact and more general function spaces.

Abstract

Let $X$ be a topological space. A subset of $C(X)$, the space of continuous real-valued functions on $X$, is a partially ordered set in the pointwise order. Suppose that $X$ and $Y$ are topological spaces, and $A(X)$ and $A(Y)$ are subsets of $C(X)$ and $C(Y)$ respectively. We consider the general problem of characterizing the order isomorphisms (order preserving bijections) between $A(X)$ and $A(Y)$. Under some general assumptions on $A(X)$ and $A(Y)$, and when $X$ and $Y$ are compact Hausdorff, it is shown that existence of an order isomorphism between $A(X)$ and $A(Y)$ gives rise to an associated homeomorphism between $X$ and $Y$. This generalizes a classical result of Kaplansky concerning linear order isomorphisms between $C(X)$ and $C(Y)$ for compact Hausdorff $X$ and $Y$. The class of near vector lattices is introduced in order to extend the result further to noncompact spaces $X$ and $Y$. The main applications lie in the case when $X$ and $Y$ are metric spaces. Looking at spaces of uniformly continuous functions, Lipschitz functions, little Lipschitz functions, spaces of differentiable functions, and the bounded, "local" and "bounded local" versions of these spaces, characterizations of when spaces of one type can be order isomorphic to spaces of another type are obtained.