On well-splitting posets

TL;DR

This paper introduces a new class of proper posets that preserve certain properties of reals in extensions and includes several well-known forcing notions, potentially advancing solutions to the Roitman problem.

Contribution

It defines a novel class of proper posets that maintain ground model reals' splitting and unboundedness, unifying various forcing notions under one framework.

Findings

Includes $oldsymbol{ extomega^oldsymbol{ extomega}}$-bounding, Cohen, Miller, and Mathias posets

Preserves ground model reals as splitting and unbounded in extensions

Provides a new approach towards solving variations of the Roitman problem

Abstract

We introduce a class of proper posets which is preserved under countable support iterations, includes $\omega^\omega$-bounding, Cohen, Miller, and Mathias posets associated to filters with the Hurewicz covering properties, and has the property that the ground model reals remain splitting and unbounded in corresponding extensions. Our results may be considered as a possible path towards solving variations of the famous Roitman problem.