Trees and gaps from a construction scheme

Abstract

We present simple constructions of trees and gaps using a general construction scheme that can be useful in constructing many other structures. As a result, we solve a natural problem about Hausdorff gaps in the quotient algebra P(ω)/Fin found in the literature. As it is well known, Hausdorff gaps can sometimes be filled in ω1-preserving forcing extensions. There are two natural conditions on Hausdorff gaps, dubbed S and T in the literature, that guarantee the existence of such forcing extensions. In part, these conditions are motivated by analogies between fillable Hausdorff gaps and Suslin trees. While the condition S is equivalent to the existence of ω1-preserving forcing extensions that fill the gap, we show here that its natural strengthening T is in fact strictly stronger.