Base Tree Property

Abstract

Building on previous work of [BPS] we investigate $\sigma$-closed partial orders of size continuum. We provide both an internal and external characterization of such partial orders by showing that (1) every $\sigma$-closed partial order of size continuum has a base tree and that (2) $\sigma$-closed forcing notions of density $\mathfrak{c}$ correspond exactly to regular suborders of the collapsing algebra $Coll({\omega }_1,2^{\omega })$. We further study some naturally ocurring examples of such partial orders.