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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.

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There is rarely need to use abbreviations in MO questions (and, in your title, calling maps with dense image surjective is strange!) –  Mariano Suárez-Alvarez Feb 23 '11 at 22:17
    
I don't think the abbreviations hinder one's ability to read the question, myself. Though I agree that the way the title is worded is momentarily confusing... –  Yemon Choi Feb 23 '11 at 23:00
    
I replaced the abbreviations - the meaning of 'cts' may not leap out at someone whose first language is not English, and 'homo' is a particularly inelegant abbreviation IMHO. –  David Roberts Feb 23 '11 at 23:18
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It's not true that a continuous homomorphism with dense range are epimorphisms, unless you work in the category of Hausdorff topological groups. This is just because you can give any group the indiscrete topology, and in that context all maps have dense image. Alternatively, you can consider the inclusion $\mathbb{Q}\to\mathbb{R}$, which equalises the projection and the zero map to the non-Hausdorff group $\mathbb{R}/\mathbb{Q}$. –  Neil Strickland Feb 24 '11 at 7:42
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Mariano is right. I would even say that a mathematical text should never contain any abbreviation. This rule has been observed, I think, by the best authors: Bourbaki (also in English) , Grothendieck, Serre , Cartan and his seminarists,... –  Georges Elencwajg Feb 24 '11 at 9:15

1 Answer 1

up vote 13 down vote accepted

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.

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It's interesting that the answer is "yes" for Hausdorff topological spaces, and "yes" for groups, but "no" for Hausdorff topological groups. –  Greg Marks Feb 23 '11 at 23:25
    
Many thanks. I guess I really should have been able to find the Nummela paper myself, given its title! –  Matthew Daws Feb 24 '11 at 8:51

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