TITLE: Epimorphisms have dense range in TopHausGrp?
QUESTION [10 upvotes]: 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.

REPLY [13 votes]: Google, MathSciNet and some ferreting lead me to

MR1235755 (94m:22003)
  Uspenskiĭ, Vladimir(D-MNCH)
  The solution of the epimorphism problem for Hausdorff topological groups.
  Sem. Sophus Lie 3 (1993), no. 1, 69–70. 

where the review indicates that the answer is negative in general, but positive for locally compact groups; this latter case was apparently treated in

MR0492044 (58 #11204)
  Nummela, Eric C.
  On epimorphisms of topological groups.
  Gen. Topology Appl. 9 (1978), no. 2, 155–167.

The case of compact groups had been done earlier by Poguntke:

MR0263978 (41 #8577)
  Poguntke, Detlev
  Epimorphisms of compact groups are onto.
  Proc. Amer. Math. Soc. 26 1970 503–504.

and this apparently inspired the authors of the following paper

MR1338245 (96c:46054)
  Hofmann, K. H.(D-DARM); Neeb, K.-H.(D-ERL-MI)
  Epimorphisms of $C^∗$-algebras are surjective.
  Arch. Math. (Basel) 65 (1995), no. 2, 134–137.