We prove results about automatic continuity and openness of abstract surjective group homomorphisms \( K\overset{\varphi }{\to }G, \) where G and K belong to a certain class К of topological groups, and where the kernel of φ satisies a certain topological countability condition. Our results apply in particular to the case where G is a semisimple Lie group or a semisimple compact group, and where К is either the class of all locally compact groups or the class of all Polish groups.

中文翻译：

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局部紧密群和波兰群之间抽象同质性的自动连续性

我们证明了关于抽象射影群同态\（K \ overset {\ varphi} {\ to} G，\）的自动连续性和开放性的结果，其中G和K属于拓扑组的某个类别К，并且φ的核满足一定的拓扑可数条件。我们的结果尤其适用于以下情况：G是半简单的李群或半简单的紧凑群，而К是所有本地紧凑群的类别或所有波兰群的类别。