# Localization of Closedness and Continuity Properties, with Applications to Preferences and Production Sets - (Now published as Localisation of continuity to bounded sets for nonmetrisable vector topologies and its applications to economic equilibrium theory, in Indagationes Mathematicae (New Series), 11 (2000), pp.53-61.)

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## Abstract

Localization techniques which facilitate verification of the topological properties of sets, functions and correspondences needed for equilibrium analysis in infinite-dimensional spaces are given. For example, it is shown that weak* upper semicontinuity (w*-u.s.c.) of a concave function (or a convex preorder) on the dual , L, of a separable Banach space, L, is equivalent to bounded w*-u.s.c. and therefore to sequential w*-u.s.c. For nondecreasing functions defined on bounded-from-below subsets of topological vector lattices, the property of lower semicontinuity can also be 'localized' to bounded regions, which in the case of L = L? with the Mackay topology leads to characterization in terms of sequences through the introduction of convergence measure. To illustrate the simplification this affords, Bewley's Mackey integration theory; some other uses are sketched or referred to.

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