We prove that groups for which every countable subgroup is free (ℵ1-free groups) are n-slender, cm-slender, and lcH-slender. In particular, every homomorphism from a completely metrizable group to an ℵ1-free group has an open kernel. We also show that ℵ1-free abelian groups are lcHslender, which is especially interesting in light of the fact that some ℵ1-free abelian groups are neither n-slender nor cm-slender. The strongly ℵ1-free abelian groups are shown to be n-slender, cm-slender, and lcH-slender. We also give a characterization of the cm-slender and lcH-slender abelian groups.

中文翻译：

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ℵ1-free 组的自动连续性

我们证明每个可数子群都是自由的群（ℵ1 自由群）是 n-slender、cm-slender 和 lcH-slender。特别是，从完全可度量化群到无ℵ1群的每个同态都有一个开放核。我们还表明无 ℵ1 的阿贝尔群是 lcHslender 的，鉴于一些无 ℵ 1 的阿贝尔群既不是 n-slender 也不是 cm-slender，这一点特别有趣。强 ℵ1-free 阿贝尔群显示为 n-slender、cm-slender 和 lcH-slender。我们还给出了 cm-slender 和 lcH-slender 阿贝尔群的表征。