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

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.