Automatic continuity of $\aleph_1$-free groups

TL;DR

This paper demonstrates that $\u22051$-free groups and abelian groups possess various slenderness properties, ensuring certain homomorphisms have open kernels, and provides characterizations of these properties in the context of abelian groups.

Contribution

It establishes that $51$-free groups are n-, cm-, and lcH-slender, and characterizes slenderness in abelian groups, expanding understanding of their algebraic and topological properties.

Findings

$51$-free groups are n-, cm-, and lcH-slender.

$51$-free abelian groups are lcH-slender.

Strongly $1f51$-free abelian groups are n-, cm-, and lcH-slender.

Abstract

We prove that groups for which every countable subgroup is free ($\aleph_1$-free groups) are n-slender, cm-slender, and lcH-slender. In particular every homomorphism from a completely metrizable group to an $\aleph_1$-free group has an open kernel. We also show that $\aleph_1$-free abelian groups are lcH-slender, which is especially interesting in light of the fact that some $\aleph_1$-free abelian groups are neither n- nor cm-slender. The strongly $\aleph_1$-free abelian groups are shown to be n-, cm-, and lcH-slender. We also give a characterization of cm- and lcH-slender abelian groups.