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

2004/07/15 by Roumen Anguelov, Anguelov, Roumen
Computer Science · Mathematics · #26E25 #35F20 #54C30 #Analysis of PDEs (math.AP) #Digital Filter Design and Implementation #FOS: Mathematics #Mathematical and Theoretical Analysis #Numerical Methods and Algorithms #math.AP #msc:26E25 #msc:35F20 #msc:54C30

paper · pdf · doi:10.48550/arxiv.math/0407272

arxiv created 2004/07/15 · openalex publication_date 2004/07/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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