When is every Hausdorff continuous interval-valued function bounded by continuous functions?

Abstract

In this paper we characterize the topological spaces X for which ℍ ft (X) = ℍ cm (X). Here ℍ ft (X) is the set of all finite Hausdorff continuous interval-valued functions on X and ℍ cm (X) is the subset of ℍ ft (X) formed by those functions f ε ℍ ft (X) for which there exist continuous real-valued functions on X, ϕ and Φ such that ϕ ≤ f ≤ Φ. These sets of interval-valued functions are important in functional analysis and in partial differential equations (PDEs) since ℍ ft (X) is Dedekind order complete and the Dedekind order completion of the set C(X) of all continuous real-valued functions on a completely regular topological space X is ℍ cm (X).