Some examples of continuous images of Radon-Nikodým compact spaces

Abstract

Abstract. We provide a characterization of continuous images of RadonNikodým compacta lying in a product of real lines and model on it a method for constructing natural examples of such continuous images.

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

“…Other articles devoted to the problem of the continuous image include [4,14,19,25]. More information can be found in [7,12,13,26], which are dedicated to the topic, or contain sections dedicated to it.…”

Compactness in Banach space theory —selected problems

“…In [3], Arvanitakis and the first author developped a method of construction of continuous images of RN compacta that are natural candidates not to be RN compact. We mention here a concrete one: the space of almost increasing functions.…”

“…Let y ∈ A, and ε = 1 − i∈κ |y i |. We can consider a sequence of elements y n ∈ A converging to x in which y n coincides with y on the support of y, and y n has a single extra nonzero coordinate with value 3 4 ε. It is possible to find a natural number k big enough so that for every (z, x) ∈ g −1 (y n ) we have j<k z j ≥ 1 − ε/2.…”

“…There has been a number of papers addressing this problem [34,38,2,16,31,33,4,7,3], including a survey article [15]. We collect in this section a list of problems scattered in the litterature and some remarks, all of them related to the continuous image problem.…”

“…The approach from [16] to quasi RN compact spaces and the recent characterization of continuous images of RN compacta in [3] suggest a relevant role of the combinatorics of N N in the problem of distinguishing quasi RN, RN compacta and their continuous images. We have also the fact that the three classes coincide for compact spaces of weight less than b [4].…”