Epimorphisms have dense range in TopHausGrp?

2011-02-23T22:15:34Z

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.

Answer by Yemon Choi for Epimorphisms have dense range in TopHausGrp?

2011-02-23T23:08:32Z

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.

Epimorphisms have dense range in TopHausGrp? - MathOverflow

Epimorphisms have dense range in TopHausGrp?

Answer by Yemon Choi for Epimorphisms have dense range in TopHausGrp?