A Selection Principle and Products in Topological Groups

TL;DR

This paper studies how certain covering properties of $T_0$ topological groups are preserved under products, finite powers, and forcing, providing new results on o-bounded groups in various set-theoretic extensions.

Contribution

It introduces new preservation results for o-bounded groups under products and forcing extensions, expanding understanding of their behavior in topological group theory.

Findings

Product of strictly o-bounded and o-bounded groups is o-bounded

Product with a ground model $ ext{aleph}_0$ bounded group remains o-bounded after forcing

Results hold in generic extensions with Hechler and Mathias reals

Abstract

We consider the preservation under products, finite powers, and forcing, of a selection principle based covering property of $T_0$ topological groups. Though the paper is in part a survey, it contributes some new information, including: 1. The product of a strictly o-bounded group with an o-bounded group is an o-bounded group - Corollary 18 2. In the generic extension by a finite support iteration of $\aleph_1$ Hechler reals the product of any o-bounded group with a ground model $\aleph_0$ bounded group is an o-bounded group - Theorem 19 3. In the generic extension by a countable support iteration of length $\aleph_2$ Mathias reals the product of any o-bounded group with a ground model $\aleph_0$ bounded group is an o-bounded group - Theorem 20.