Dedekind order completion of C(X) by Hausdorff continuous functions

TL;DR

This paper extends Hausdorff continuous functions to topological spaces and proves they form a Dedekind order complete set, solving a long-standing open problem and impacting nonlinear PDE solutions.

Contribution

It demonstrates that finite Hausdorff continuous functions form a Dedekind order complete set, including the completions of C(X) and C_b(X), and characterizes these completions within Hausdorff continuous functions.

Findings

Finite Hausdorff continuous functions are Dedekind order complete.

The Dedekind order completions of C(X) and C_b(X) are characterized within Hausdorff continuous functions.

Solves an open problem and has applications to nonlinear PDEs.

Abstract

The concept of Hausdorff continuous interval valued functions, developed within the theory of Hausdorff approximations and originaly defined for interval valued functions of one real variable is extended to interval valued functions defined on a topological space X. The main result is that the set of all finite Hausdorff continuous functions on any topological space X is Dedekind order complete. Hence it contains the Dedekind order completion of the set C(X) of all continuous real functions defined on X as well as the Dedekind order completion of the set C_b(X) of all bounded continuous functions on X. Under some general assumptions about the topological space X the Dedekind order completions of both C(X) and C_b(X) are characterised as subsets of the set of all Hausdorff continuous functions. This solves a long outstanding open problem about the Dedekind order completion of C(X). In addition, it has major applications to the regularity of solutions of large classes of nonlinear PDEs.