A Strong Kurepa Tree

TL;DR

This paper demonstrates the consistency of a Kurepa tree whose branch space maps continuously onto the Cantor space, solving an open problem and revealing structural properties of such trees.

Contribution

It proves the existence of a Kurepa tree with a continuous image of its branches onto the Cantor space, addressing an open problem in set theory.

Findings

Existence of a Kurepa tree with a continuous image of branches onto the Cantor space.

Such a Kurepa tree contains an Aronszajn subtree.

The result resolves an open problem in the field.

Abstract

We prove that it is consistent that there exists a Kurepa tree $T$ such that ${}^{\omega_1}2$ is a continuous image of the topological space $[T]$ consisting of all cofinal branches of $T$ with respect to the cone topologies. This result solves an open problem due to Bergfalk, Chodounsk\'{y}, Guzm\'{a}n, and Hru\v{s}\'{a}k. We also prove that any Kurepa tree with the above property contains an Aronszajn subtree.