Abstract
In this paper we discuss the algebraic structure of the space H(X) of finite Hausdorff continuous interval functions defined on an arbitrary topological space X. In particular, we show that H(X) is a linear space over R containing C(X), the space of continuous real functions on X, as a linear subspace. In addition, we prove that the order on H(X) is compatible with the linear structure introduced here so that H(X) is an Archimedean vector lattice.
Highlights
Interval functions, and interval analysis in general, is typically associated with numerical analysis and validated computing, see for instance [22], [23]
Our method reveals the essential mathematical mechanism that allows the extension of the linear operations on C (X) to the larger set H(X), namely, a minimality condition satisfied by H-continuous functions which is preserved by pointwise interval arithmetic [3]
J H van der Walt, The Linear Space of Hausdorff Continuous Interval Functions gives some direction as to how one may proceed in the case of functions defined on a general topological space
Summary
Interval analysis in general, is typically associated with numerical analysis and validated computing, see for instance [22], [23]. As pointed out in [21], biological dynamic systems typically involve uncertain data and / or parameters, numerical and / or inherent sensitivity and structural uncertainties which necessitate model validation Problems related to these issues of uncertainty and sensitivity, including computing enclosures for sets of solutions [24] and estimation of parameter ranges [15], essentially belong to Set-Valued Analysis in general and are often addressed within the setting of Interval Analysis. Our method gives an indication of how to define algebraic operations on certain spaces of setvalued maps that take values in a general metrisable topological vector space This problem is addressed in [7] in the case when the domain is a Baire space, while the method developed here for H-continuous functions. J H van der Walt, The Linear Space of Hausdorff Continuous Interval Functions gives some direction as to how one may proceed in the case of functions defined on a general topological space
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