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.



Epimorphisms have dense range in TopHausGrp?
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.](http://dx.doi.org/10.1016/0016-660X%2878%2990060-0)
> 
> 
> 


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. 
> 
> 
> 


