Internal characterization of fragmentable spaces. (English) Zbl 0645.46017

Let X be a topological space and let \(\rho\) be a metric on \(X\times X\). X is said to be fragmented by the metric \(\rho\) if for every \(\epsilon >0\) and each nonempty subset Y of X there is a nonempty relatively open subset U of Y such that \(\rho\)-diam(U)\(\leq \epsilon\) [J. E. Jayne and C. A. Rogers, Acta Math. 155, 41-79 (1985; Zbl 0588.54020)]. The author shows that X is fragmentable if and only if X admits a separating \(\sigma\)-relatively open partitioning. This characterisation is used to prove the main result (Theorem 3.1): Let X be a Hausdorff compact space which is fragmented by a metric. Then \(C(X)^*\) endowed with the weak star topology is a framentable space. Consequently (Corollary 3.6) C(X) is a weak Asplund space.
Reviewer: R.Cross


46B20 Geometry and structure of normed linear spaces
54E35 Metric spaces, metrizability
46B25 Classical Banach spaces in the general theory


Zbl 0588.54020
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