Rendiconti del Seminario Matematico della Università di Padova
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Published online: 2008-12-31
Semitopological HomomorphismsAnna Giordano Bruno (1) Università di Udine, Italy
Inspired by an analogous result of Arnautov about isomorphisms, we prove that all continuous surjective homomorphisms of topological groups f : G → H can be obtained as restrictions of open continuous surjective homomorphisms f˜ : G˜ → H, where G is a topological subgroup of G˜. In case the topologies on G and H are Hausdorff and H is complete, we characterize continuous surjective homomorphisms f : G → H when G has to be a dense normal subgroup of G˜.
We consider the general case when G is requested to be a normal subgroup of G˜, that is when f is semitopological — Arnautov gave a characterization of semitopological isomorphisms internal to the groups G and H. In the case of homomorphisms we define new properties and consider particular cases in order to give similar internal conditions which are sufficient or necessary for f to be semitopological. Finally we establish a lot of stability properties of the class of all semitopological homomorphisms.
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Giordano Bruno Anna: Semitopological Homomorphisms. Rend. Sem. Mat. Univ. Padova 120 (2008), 79-126. doi: 10.4171/RSMUP/120-6