$\sigma$-continuity and related forcings

TL;DR

This paper characterizes Steprans forcing through trees, explores its properties like fusion and continuous reading of names, and connects it to Miller forcing, revealing new insights into related forcing notions.

Contribution

It provides a tree-based characterization of Steprans forcing and links it to Miller forcing, highlighting differences in continuous reading of names among various forcings.

Findings

Steprans forcing characterized via trees

Fusion property established for Steprans forcing

Connection between Steprans and Miller forcing

Abstract

The Steprans forcing notion arises as a quotient of Borel sets modulo the ideal of $\sigma$-continuity of a certain Borel not $\sigma$-continuous function. We give a characterization of this forcing in the language of trees and using this characterization we establish such properties of the forcing as fusion and continuous reading of names. Although the latter property is usually implied by the fact that the associated ideal is generated by closed sets, we show it is not the case with Steprans forcing. We also establish a connection between Steprans forcing and Miller forcing thus giving a new description of the latter. Eventually, we exhibit a variety of forcing notions which do not have continuous reading of names in any presentation.