Closure properties of parametric subcompleteness

Abstract

For an ordinal $\varepsilon$, I introduce a variant of the notion of subcompleteness of a forcing poset, which I call $\varepsilon$-subcompleteness, and show that this class of forcings enjoys some closure properties that the original class of subcomplete forcings does not seem to have: factors of $\varepsilon$-subcomplete forcings are $\varepsilon$-subcomplete, and if $\mathbb {P}$ and $\mathbb {Q}$ are forcing-equivalent notions, then $\mathbb {P}$ is $\varepsilon$-subcomplete iff $\mathbb {Q}$ is. I formulate a Two Step Theorem for $\varepsilon$-subcompleteness and prove an RCS iteration theorem for $\varepsilon$-subcompleteness which is slightly less restrictive than the original one, in that its formulation is more careful about the amount of collapsing necessary. Finally, I show that an adequate degree of $\varepsilon$-subcompleteness follows from the $\kappa$-distributivity of a forcing, for $\kappa >\omega _1$.