Logical opens of exponential objects

Abstract

Let X = C^∞(R^p)/J, Y = C^∞(R^n)/I be two representable objects in the Dubuc topos D (see Secsion 0) where J has line determined extensions (0.3). The main result in this paper (Theorem 1.11) says that the global section functor Γ establishes a bijection between Penon open sub-objects of Y^x and open subsets of Γ(Y^x) in the C^∞-CO topology. We show also that when I = {0}, we can assume J arbitrary (1.12). However, the restriction on J (of having line determined extensions) is seen to be unavoidable in general. We precede the article with a Section 0 where we recall all these notions and fix the notations.

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