A topic of continued interest in set-theoretic topology is the question of which topological properties are preserved under which forcings. In recent work, the speaker and Holshouser have shown that strongly proper forcings preserve a wide variety of covering properties (including Lindelöf), generalizing work of Dow, Iwasa, and Kada. In this talk, we will give an overview of yet further work done by the speaker on this topic. Namely, we discuss how to create forcing axioms for proper posets that preserve a given Lindelöf space. This uses Neeman’s two-type side conditions machinery in combination with the earlier work of Gilton and Holshouser.