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On Cantor’s intersection theorem in \(C(K)\) spaces. (English) Zbl 1420.52001

This paper deals with Cantor’s intersection theorem in \(C(K)\) – the Banach space of continuous real functions defined on a compact Hausdorff space \(K\), endowed with the supremum norm. The first part contains a characterization of finite dimension of \(C(K)\) spaces (i.e., finite \(K\)) in terms of the validity of Cantor’s intersection theorem: every nested family of intervals has a nonempty intersection, where an interval \([f,g]\) in \(C(K)\) is the set \(\left\{h \in C(K) : f \le h \le g\right\}\) considering \(f\) and \(g\) bounded real functions on \(K\), not necessarily belonging to \(C(K)\). If intervals are defined taking the bounding functions continuous, the corresponding version of Cantor’s intersection theorem characterizes injectivity of \(C(K)\) spaces. The last part of the paper studies additional properties of nested families of intervals in \(C(K)\) that yield their nonempty intersection. These conditions are related to equicontinuity and diametrical maximality.

MSC:

52A05 Convex sets without dimension restrictions (aspects of convex geometry)
46J10 Banach algebras of continuous functions, function algebras
46E15 Banach spaces of continuous, differentiable or analytic functions
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