o-Boundedness of free objects over a Tychonoff space

TL;DR

This paper characterizes the o-boundedness of free topological groups and locally-convex spaces over Tychonoff spaces, linking it to selection principles and invariance under certain set-theoretic assumptions.

Contribution

It provides a characterization of o-boundedness for free topological structures in terms of properties of the underlying space, connecting to selection principles and invariance results.

Findings

o-boundedness characterized via properties of Tychonoff spaces

Hurewicz and Menger properties are l-invariant under ZFC

Construction of topological groups with strong combinatorial properties

Abstract

In this paper we characterize [strict] o-boundedness of the free (abelian) topological group F(X) (A(X)) as well as the free locally-convex linear topological space L(X) in terms of properties of a Tychonoff space X. These properties appear to be close to so-called selection principles, which permits us to show, that (it is consistent with ZFC that) the property of Hurewicz (Menger) is l-invariant. This gives a method of construction of OF-undetermined topological groups with strong combinatorial properties.