  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.

Group Theory