A guessing principle from a Souslin tree, with applications to topology

TL;DR

This paper introduces a new combinatorial principle, $\uclubsuit_{AD}$, derived from Souslin trees, and applies it to construct specific topological spaces, providing new insights into their existence and properties.

Contribution

It establishes a link between Souslin trees and the combinatorial principle $\uclubsuit_{AD}$, enabling new topological space constructions and simplifying existing proofs.

Findings

Strong $\uclubsuit_{AD}$ follows from the existence of a Souslin tree.

Weak $\uclubsuit_{AD}$ does not follow from an almost Souslin tree.

A simplified proof that Souslin trees imply the existence of a Dowker $S$-space.

Abstract

We introduce a new combinatorial principle which we call $\clubsuit_{AD}$. This principle asserts the existence of a certain multi-ladder system with guessing and almost-disjointness features, and is shown to be sufficient for carrying out de Caux type constructions of topological spaces. Our main result states that strong instances of $\clubsuit_{AD}$ follow from the existence of a Souslin tree. It is also shown that the weakest instance of $\clubsuit_{AD}$ does not follow from the existence of an almost Souslin tree. As an application, we obtain a simple, de Caux type proof of Rudin's result that if there is a Souslin tree, then there is an $S$-space which is Dowker.