Kurepa trees, continuous images, and perfect set properties

TL;DR

This paper explores the properties of higher Kurepa trees using descriptive set theory, demonstrating the consistency of certain set-theoretic configurations and analyzing perfect set properties and representations of branches.

Contribution

It proves the consistency of the existence of -Kurepa trees with non-continuous branch sets under CH and investigates the necessity of Kurepa trees for certain set properties.

Findings

Existence of -Kurepa trees with non-continuous branch sets is consistent with CH.

Kurepa trees are not necessary for certain closed sets to fail perfect set properties.

Established models for full and superthin trees with specific properties.

Abstract

Building upon work of L\"{u}cke and Schlicht, we study (higher) Kurepa trees through the lens of higher descriptive set theory, focusing in particular on various perfect set properties and representations of sets of branches through trees as continuous images of function spaces. Answering a question of L\"{u}cke and Schlicht, we prove that it is consistent with $\mathsf{CH}$ that there exist $\omega_2$-Kurepa trees and yet, for every $\omega_2$-Kurepa tree $T \subseteq {^{<\omega_2}}\omega_2$, the set $[T] \subseteq {^{\omega_2}}\omega_2$ of cofinal branches through $T$ is not a continuous image of ${^{\omega_2}}\omega_2$. We also produce models indicating that the existence of Kurepa trees is not necessary to produce closed subsets of ${^{\omega_1}}\omega_1$ failing to satisfy strong perfect set properties, and prove a number of consistency results regarding \emph{full} and \emph{superthin} trees.