A small Radon-Nikodým compact space from a parametrized diamond
Arturo Martínez-Celis, Adam Morawski
TL;DR
The paper investigates whether Radon-Nikodým compact spaces are closed under continuous images and, under the combinatorial principle $\diamondsuit(\mathrm{non}(\mathcal{M}))$, constructs a Radon-Nikodým compact space of weight $\aleph_1$ whose continuous image is not Radon-Nikodým. It adapts Avilés–Koszmider’s basic space framework to a refined basic$^*$ setting and uses a parametrized diamond (via a Borel encoding and a diamond sequence) to build a small basic$^*$ space with carefully arranged almost-disjoint families. This yields a pair of spaces $\mathbb{L}_0$ and $\mathbb{L}_1$ with $\mathbb{L}_1$ a continuous image of $\mathbb{L}_0$, where $\mathbb{L}_0$ is Radon-Nikodým and $\mathbb{L}_1$ is not, thereby showing the consistency of a RN compact space of small weight having a non-RN image. The work also highlights inherent weight limitations of the method (e.g., $w(K)\ge\mathrm{non}(\mathcal{M})$) and points to possible refinements (such as $\diamondsuit(\mathfrak{b})$) to explore different weight regimes.
Abstract
A compact space $K$ is Radon-Nikodým if there is a lower semi-continuous metric fragmenting $K$. In this note, we show that, under $\diamondsuit (\mathrm{non}{\mathcal{M}})$, there is a Radon-Nikodým compact space of weight $\aleph_1$ with a continuous image that is not Radon-Nikodým, which partially answers a question posed in arXiv:1112.4152 [math.FA].