$\mathsf{PFA}$ and $\omega_1$-free compact spaces

TL;DR

This paper explores the implications of the Proper Forcing Axiom on compact Hausdorff spaces, showing they are either first-countable or contain a converging sequence of length 1.

Contribution

It establishes a dichotomy for compact Hausdorff spaces under PFA, linking set-theoretic axioms to topological properties.

Findings

Under PFA, compact Hausdorff spaces are either first-countable or contain a converging 1-sequence.

The paper discusses the role of 1-free compact spaces.

It provides conditions under which certain topological spaces exhibit these properties.

Abstract

The Proper Forcing Axiom implies that compact Hausdorff spaces are either first-countable or contain a converging $\omega_1$-sequence.