## 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

### MSC:

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

### Keywords:

weak star topology; framentable space; Asplund space

Zbl 0588.54020
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### References:

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