Rendiconti del Seminario Matematico della Università di Padova

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Volume 120, 2008, pp. 79–126
DOI: 10.4171/RSMUP/120-6

Published online: 2008-12-31

Semitopological Homomorphisms

Anna Giordano Bruno[1]

(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 : GH 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 : GH 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