Forcing Axioms for Proper Posets Preserving a Topological Property: Consistency Results

TL;DR

This paper introduces two new forcing axioms related to proper posets that preserve certain topological properties, using Neeman's iteration schema to establish their consistency, with future work planned on their applications.

Contribution

The paper presents two novel forcing axioms for proper posets that maintain Lindelöf and countably tight properties, proven consistent via Neeman's side conditions iteration.

Findings

Established the consistency of the new forcing axioms.

Connected the axioms to preservation of topological properties.

Laid groundwork for future applications of these axioms.

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{\"o}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.