RADON-NIKODÝM COMPACT SPACES AND BANACH SPACES OF LOW WEIGHT

ANTONIO AVILÉS

The concept of Radon-Nikodým compact space has its origin in Banach space theory, and it is defined as a topological space which is homeomorphic to a weak^{∗} compact subset of the dual of an Asplund space (that is, a dual Banach space with the Radon-Nikodým property). It is an open problem whether a continuous image of a Radon-Nikodým compact is a Radon-Nikodým compact. We present a new partial answer to this problem: continuous images of

Radon-Nikodým compacta of weight less than cardinal b are Radon-Nikodým.

Cardinal b is the least cardinal of a subset ω^{ω} not contained in a σ-compact subset. It is consistent that b > ω_{1} and the statement b = c is weaker than Martin’s axiom.

This result on continuous images has a Banach space counterpart: If a Banach space of density character less than b is a subspace of an Asplund generated space (respectively a subspace of a WCG space), then it is Asplund generated (respectively WCG). There are counterexamples to this statements of density character exactly b.

Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo (Murcia), Spain

   E-mail address: avileslo@um.es