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Nonlinear order isomorphisms on function spaces

Volume 517 / 2016

Denny H. Leung, Wee-Kee Tang Dissertationes Mathematicae 517 (2016), 1-75 MSC: Primary 06F20, 46E05, 47H07, 54C30, 54F05. DOI: 10.4064/dm737-11-2015 Published online: 11 July 2016

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, Lipschitz, little Lipschitz and differentiable functions, and the bounded, “local” and “bounded local” versions of these spaces, criteria of when spaces of one type can be order isomorphic to spaces of another type are obtained.

Authors

  • Denny H. LeungDepartment of Mathematics
    National University of Singapore
    Singapore 119076
    e-mail
  • Wee-Kee TangDivision of Mathematical Sciences
    School of Physical and Mathematical Sciences
    Nanyang Technological University
    Singapore 637371
    e-mail

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