The Lindelöf property and fragmentability
by
B. Cascales
Departamento de Matematicas. Universidad de Murcia. Spain
Coauthors: I. Namioka (University of Washington), G. Vera (Universidad de Murcia)

The notion of fragmentability as stated below was introduced by Jayne and Rogers [9].

Definition.- Let (X, \tau) be a topological space and \rho a metric on X. We say that (X, \tau) is fragmented by \rho (or \rho-fragmented) if for each non-empty subset A of X and for each \epsilon > 0 there exists a non-empty \tau-open subset U of X such that U \cap A =/= \emptyset and \rho- diam (U \cap A) <= \epsilon.

It is a classical result by I. Namioka, [10], that if C(K) is the space of continuous functions on a compact space K, the pointwise compact subsets of C(K) are fragmented by the supremum norm. Compact sets fragmented by a lower semicontinuous metric are called Radon-Nikodým compact and they are homeomorphic to a weak ^* compact subset of a dual Banach space with the Radon-Nikodým Property (RNP), [11]. We will present the following result:

Theorem.- Let K be a compact space, D a dense subset of K, t_p(D) (resp. t_p(K)) the topology of pointwise convergence on D (resp. on K). Then, every t_p(D)-compact subset of C(K) which is t_p(K)-Lindelöf is fragmented by the supremum norm, and so, it is a Radon-Nikodym compact space.

This result solves a problem of [5] and it is a common generalization of the result stating the fragmentability of pointwise compact subsets in spaces C(K) and the result in [2] saying that convex weak ^* compact weakly Lindelöf subsets of dual Banach spaces do have the RNP (in particular weakly Lindelöf dual Banach spaces have RNP, [8]). The result presented here is very much related to the problem of knowing if l^\infty=C(\betaN) contains a t_p(\betaN)-Lindelöf subset Y separating the points of \betaN, see [1, p. 610] for a related open problem. We prove that this is impossible when Y is assumed to be t_p(N)-Cech-analytic. Several other non trivial applications of this theorem will be presented.

We will also comment on how it is possible to prove that when K is a compact subset of the cube [0, 1]^D then the fragmentability of K by the metric

d(x, y)= sup {|x(t)-y(t)|: t in D},     for   x, y in [0, 1]D

is equivalent to the fact of K being Lindelöf for the topology of uniform convergence on sequences on D. As a consequence of this result we get that a dual Banach space X ^* has the RNP if, and only if, X ^* is Lindelöf for the topology of uniform convergence on bounded sequences of X, [12]. A combination of the previous statements and results by Edgar and Pol will allow us to prove that if a dual Banach space X ^* is weakly Lindelöf, then (X ^* , weak)ⁿ is Lindelöf for every n in N, what partially answers a question by Corson, [12].

References

1. A.V. Arkhangleskii. Open Problems in Topology, chapter: Problems in C_p-theory, pages 601-617. North-Holland, 1990.
2. R. D. Bourgin. Geometric aspects of convex sets with the Radon-Nikodym property. LNM. Springer-Verlag, 993, 1983.
3. B. Cascales and G. Godefroy. Angelicity and the boundary problem. Mathematika, 45(1):105-112, 1998.
4. B. Cascales, G. Manjabacas, and G. Vera. A Krein-Smulian type result in Banach spaces. Oxford Quarterly Journal Math, 48(2):161-167, 1997.
5. B. Cascales, G. Manjabacas, and G. Vera. Fragmentability and compactness in C(K)-spaces. Stud. Math., 131 (1):73-87, 1998.
6. B. Cascales, E. Matouskova, I. Namioka, and J. Orihuela. Continuous images of RN compact spaces and the Lindelöf property. Work in progress.
7. B. Cascales, I. Namioka, and G. Vera. The Lindelöf property and fragmentability. To appear P.A.M.S. 11 pages, 2000.
8. G. A. Edgar. Measurability in a Banach space I. Indiana Univ. Math. J., 26:663-677, 1977.
9. J. E. Jayne and C. A. Rogers. Borel selectors for upper semicontinuous set-valued maps. Acta Math., 155:41-79, 1985.
10. I. Namioka. Separate continuity and joint continuity. Pac. J. Math., 51(2):515-531, 1974.
11. I. Namioka. Radon-Nikodym compact spaces and fragmentability. Mathematika, 34:258-281, 1989.
12. J. Orihuela. On weakly Lindelöf Banach spaces. Progress in Functional Analysis Math.Studies North Holland, 170:279-291, 1992.

Date received: June 14, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-21.