Epimorphisms have dense range in TopHausGrp? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T09:38:48Z http://mathoverflow.net/feeds/question/56453 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56453/epimorphisms-have-dense-range-in-tophausgrp Epimorphisms have dense range in TopHausGrp? Matthew Daws 2011-02-23T22:15:34Z 2011-02-24T08:58:31Z <p>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?</p> <p>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 <a href="http://www.jstor.org/pss/2317336" rel="nofollow">http://www.jstor.org/pss/2317336</a> It is far from obvious to me that this argument extends to the topological case (but perhaps it does).</p> <p><strong>Edit:</strong> As suggested in the comments, I really was to ask about <em>Hausdorff</em> topologies.</p> http://mathoverflow.net/questions/56453/epimorphisms-have-dense-range-in-tophausgrp/56459#56459 Answer by Yemon Choi for Epimorphisms have dense range in TopHausGrp? Yemon Choi 2011-02-23T23:08:32Z 2011-02-23T23:08:32Z <p>Google, MathSciNet and some ferreting lead me to</p> <blockquote> <p>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. </p> </blockquote> <p>where the review indicates that the answer is negative in general, but positive for locally compact groups; this latter case was apparently treated in</p> <blockquote> <p>MR0492044 (58 #11204) Nummela, Eric C. On epimorphisms of topological groups. <a href="http://dx.doi.org/10.1016/0016-660X%2878%2990060-0" rel="nofollow">Gen. Topology Appl. 9 (1978), no. 2, 155–167.</a></p> </blockquote> <p>The case of compact groups had been done earlier by Poguntke:</p> <blockquote> <p>MR0263978 (41 #8577) Poguntke, Detlev Epimorphisms of compact groups are onto. Proc. Amer. Math. Soc. 26 1970 503–504.</p> </blockquote> <p>and this apparently inspired the authors of the following paper</p> <blockquote> <p>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. </p> </blockquote>