Nonlinear order isomorphisms on 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.

Order isomorphisms between bases of topologies

“…Anyhow, the papers where we find some of the most accurate results about isomorphisms of spaces of uniformly continuous functions [5,6], Lipschitz functions [4] and smooth functions [3] have something in common: the result labelled in the present paper as Lemma 2.1; the interested reader should take a close look at [21], where the authors were able to unify all these results and find new ones. This lemma has been key in these works, and has recently lead to similar results, see [7,9].…”

Nonlinear biseparating maps

“…The characterization theorem applies in particular to a number of familiar (vector-valued) function spaces such as spaces of continuous, uniformly continuous, Lipschitz and differentiable functions. We would like to point out a general resemblance of Theorem 2.8 with the fundamental characterization theorem for "nonlinear order isomorphisms" [30,Theorem 2.11]. Indeed, our study of nonlinear biseparating maps is motivated and informed by the study of nonlinear order isomorphisms at various points.…”

“…See the paragraph preceding Lemma 2.3. The theory of nonlinear biseparating maps is somewhat related to the theory or order isomorphisms developed in [30]. It also partly generalizes the notion of "nonlinear superposition operators".…”

“…However, the lack of an order makes many of the arguments more difficult in the present case, especially for uniformly continuous and Lipschitz functions. For further information on nonlinear order isomorphisms on function spaces, we refer to [30] and the references therein.…”

“…[30, Lemma 3.2] Let Y be a realcompact space and let y 0 ∈ βY \Y . There exist open sets U n and V n in βY , n ∈ N, such that (1) U n βY ⊆ V n for all n;…”