We prove results about automatic continuity and openness of abstract surjective group homomorphisms K ϕ − − → G, where G and K belong to a certain class K of topological groups, and where the kernel of ϕ satisfies 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 K is either the class of all locally compact groups or the class of all Polish groups.