Base Tree Property

Building on previous work of [BPS] we investigate $σ$-closed partial orders of size continuum. We provide both an internal and external characterization of such partial orders by showing that (1) every $σ$-closed partial order of size continuum has a base tree and that (2) $σ$-closed forcing notions of density $\mathfrak c$ correspond exactly to regular suborders of the collapsing algebra $Coll(ω_1, 2^ω)$. We further study some naturally ocurring examples of such partial orders.

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