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 $\mathcal {P}(\omega )/\textrm {Fin}$ found in the literature. As it is well known, Hausdorff gaps can sometimes be filled in $\omega _1$-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 $\omega _1$-preserving forcing extensions that fill the gap, we show here that its natural strengthening $T$ is in fact strictly stronger.