The epimorphism problem for Hausdorff topological groups

AbstractGiven a Hausdorff topological group G and a proper closed subgroup H, must there exist two distinct morphisms f, g : G → K of Hausdorff groups with ƒ | H = g | H? Equivalently, must epimorphisms in the category of Hausdorff topological groups have dense ranges? This question, asked by K.H. Hofmann in the sixties, appears as problem 512 in “Open Problems in Topology”. We answer the question in the negative