Rosenthal compacta that are premetric of finite degree

TL;DR

This paper characterizes certain separable Rosenthal compacta that are finite-degree preimages of metric compacta, showing they contain specific complex substructures, thus generalizing previous results for the case n=2.

Contribution

It extends the classification of Rosenthal compacta by identifying the presence of specific substructures based on their finite preimage degree, generalizing earlier work for n=2.

Findings

Contains a closed subset homeomorphic to the n-Split interval or the Alexandroff n-duplicate.

Generalizes previous results from n=2 to arbitrary finite n.

Provides structural insights into the topology of Rosenthal compacta.

Abstract

We show that if a separable Rosenthal compactum $K$ is an $n$-to-one preimage of a metric compactum, but it is not an $n-1$-to-one preimage, then $K$ contains a closed subset homeomorphic to either the $n-$Split interval $S_n(I)$ or the Alexandroff $n-$plicate $D_n(2^\mathbb{N})$. This generalizes a result of the third author that corresponds to the case $n=2$.