Consider the category of Topological Groups with continuous homomorphisms. Then a continuous homomorphism $f:G\rightarrow H$ with dense range is an epimorphism. Is the converse true? If not, what about for locally compact groups?
Even for groups, without topology, this is not trivial-- Wikipedia points me to a simple proof given by Linderholm, "A Group Epimorphism is Surjective", The American Mathematical Monthly Vol. 77, No. 2 (Feb., 1970), pp. 176-177 see http://www.jstor.org/pss/2317336 It is far from obvious to me that this argument extends to the topological case (but perhaps it does).
Edit: As suggested in the comments, I really was to ask about Hausdorff topologies.