§0. Introduction. J. E. Jayne and C. A. Rogers in [7] introduced the following notion.Definition. Let X be a topological space and p be a metric defined on X x X. X is said to be fragmented by the metric p if, for every e > 0 and each nonempty subset Y of X there is a nonempty relatively open subset U of Y such thatThis notion turned out to be useful in the questions concerned with the existence of "nice" selectors for upper semi-continuous compact-valued maps (see Jayne and Rogers [7] and Hansell, Jayne and Talagrand [6]). The following result is a consequence of the proof of Theorem 4 in [7] and is contained in Theorem 1' of [6]. THEOREM.Let F : X -» Y be an upper semicontinuous map with nonempty compact values from the metric space X into the fragmented Hausdorff topological space Y. Then there exists a selector f for F which is a-discrete and Borel class 1 with respect to the topology in Y, and the set of points of discontinuity of f is a set of first category in X.The conclusion of this theorem doesn't depend on a fixed metric which fragments Y but only on the existence of such a metric. We will be interested in the following class of spaces:Definition. The topological space X is said to be fragmentable if there exists a metric on X x X which fragments X. •In [12] Namioka investigated a class of Hausdorff compact spaces called [ Radon-Nikodym compacta. A Hausdorff compact space is said to be a Radon-Nikodym compact if it is homeomorph to a weak star compact subspace of a Banach space which is conjugate to an Asplund space. This class is larger than the class of Eberlein compacta and has many nice properties. All RadonNikodym compacta are fragmented by a dual norm in some conjugate Banach space.In this paper we give a necessary and sufficient condition for a topological space to be a fragmentable one. This condition allows us to prove that topological spaces which admit cr-distributively point-finite T 0 -separating family of open subsets (in particular all Talagrand and Gul'ko compacta, see \ [5,13]) are fragmentable spaces. Using this characterization we prove that if ; X is a fragmentable Hausdorff compact space the space C(X)* endowed with [MATHEMATIKA, 34 (1987), [243][244][245][246][247][248][249][250][251][252][253][254][255][256][257]

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Abstract.The techniques of Preiss-Phelps-Namioka is used to prove that if a Banach space E admits a Gâteaux smooth norm then its dual E* , endowed the weak star topology, is fragmentable. IntroductionThe main aim of this paper is to prove the title statement and to establish further relations between the theory of weak Asplund spaces and the notion of fragmentability. The latter notion was introduced by Jayne and Rogers in [JR]. 0.1. Definition. Let X be a topological space and p be a metric on it. We say that p fragments X if for every e > 0 and for every nonempty subset Y of X there exists a nonempty relatively open subset U of Y whose /A-diameter is not greater than e. The topological space X is said to be fragmentable if there exists a fragmenting metric on it. 0.2. Definition. The Banach space E is called weak Asplund if every convex function defined on an open convex subset of E is Gâteaux differentiable at each point of a dense G¿ subset of its domain.For additional information about weak Asplund spaces the reader is referred to the book of R. Phelps [Ph]. 0.3. Theorem of Asplund. If the dual space E* admits a dual strictly convex norm then E is weak Asplund.