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

We prove that groups for which every countable subgroup is free (${\mathrm{\aleph }}_1$-free groups) are n-slender, cm-slender, and lcH-slender. In particular every homomorphism from a completely metrizable group to an ${\mathrm{\aleph }}_1$-free group has an open kernel. We also show that ${\mathrm{\aleph }}_1$-free abelian groups are lcH-slender, which is especially interesting in light of the fact that some ${\mathrm{\aleph }}_1$-free abelian groups are neither n- nor cm-slender. The strongly ${\mathrm{\aleph }}_1$-free abelian groups are shown to be n-, cm-, and lcH-slender. We also give a characterization of cm- and lcH-slender abelian groups.