Forcing Axioms for Proper Posets Preserving a Topological Property: Consistency Results
Thomas Gilton
TL;DR
This work defines forcing axioms $\mathsf{PFA}_\Phi(X)$ for topological spaces $X$ satisfying a property $\Phi$, focusing on Lindelöf and countably tight cases. It develops a Neeman-style finite-support side-condition forcing $\mathbb{P}_\Phi(X)$ and proves key embedding and preservation lemmas, establishing the consistency of $\mathsf{PFA}_L(X)$ and $\mathsf{PFA}_{CT}(X)$ under a ground model with GCH and a supercompact cardinal (via a Laver function). The main contributions include the model-sequence poset framework, dense-embedding results, and the side-conditions augmentation technique showing preservation of $\Phi$ through the forcing, enabling targeted consequences while controlling $\omega_1$ and higher cardinals. These results pave the way for applications of topology-preserving forcing axioms and motivate open questions about the spectrum of possible consequences as $X$ and $\Phi$ vary.
Abstract
Forcing axioms are generalizations of Baire category principles that allow one to intersect more dense open sets and to do so in a wider variety of circumstances. In this paper we introduce two new forcing axioms related to posets which preserve topological properties of various spaces, specifically the properties of Lindel{ö}f and countably tight. The focus in this paper is on using Neeman's side conditions iteration schema to prove the consistency of these two forcing axioms. In later work, we will discuss applications of these forcing axioms.