Automatic continuity of ${\aleph }_1$-free groups

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.