A continuous image of a Radon–Nikodým compact space which is not Radon–Nikodým

Abstract

We construct a continuous image of a Radon-Nikodým compact space which is not Radon-Nikodým compact, solving the problem posed in the 80ties by Isaac Namioka.

On the problem of compact totally disconnected reflection of nonmetrizability

“…However, our results concerning Question 1.1 (2) show that attacking this question with split compact spaces as in Definition 2.1 is not optimal in the sense that taking this way it turns out that we end up facing a well-known and apparently harder problem whether locally connected perfectly normal compact spaces must be metrizable (see e.g. [31]).…”

“…To obtain counterexamples to the second question from the above examples one needs to do a bit more work. We review these and other examples in the context of Question 1.1 (2) in Proposition 4.2.…”

“…It can be sometimes used to replace an application of ♣ by a ZFC argument. Its more complicated versions were used by A. Dow and coauthors in Examples 2.15 and 2.16 of [12] others were used in Section 5 of [2], however, we feel that despite their simplicity these principles are not widely known.…”

“…To avoid repetitions we decided to present the results in the generality of split compact spaces (2.1) which allows to rely heavily in (c) and (d) on the results of K. Kunen from [31] concerning Fillipov's spaces. As counterexamples to Question 1.1 (2), to survive the impact of MA+¬CH, must be hereditarily Lindelöf and hereditarily separable (see 4.2) one perhaps could consider connected and ZFC versions of constructions like in [27] and [5] for which Kunen's OCA argument does not apply and the spaces are still preimages of metric spaces with metrizable fibers.…”

“…Despite some progress in this direction, e.g. showing it for arbitrary compact topological groups ( [36]) it turned out only recently that there are C(K)s not isomorphic to C(L)s or L totally disconnected ( [25], for further references see [28]) and that such Ks can be relatively nice like in [2]. Hence we cannot assume in the isomorphic theory that all Banach spaces C(K) are given by totally disconnected compact Ks.…”