Arnautov's problems on semitopological isomorphisms

Dikran Dikranjan, Anna Giordano Bruno


Semitopological isomorphisms of topological groups were introduced by Arnautov [2], who posed several questions related to compositions of semitopological isomorphisms and about the groups G (we call them Arnautov groups) such that for every group topology on G every semitopological isomorphism with domain (G, ) is necessarily open (i.e., a topological isomorphism). We propose a different approach to these problems by introducing appropriate new notions, necessary for a deeper understanding of Arnautov groups. This allows us to find some partial answers and many examples. In particular, we discuss the relation with minimal groups and non-topologizable groups.


A-complete topology; Heisenberg group; Markov group; Minimal group; Open mapping theorem; Permutations group; Semitopological isomorphism; Taımanov topology; Topologizable group

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V. I. Arnautov, Semitopological isomorphisms of topological rings (Russian), Mathematical Investigations (1969) 4:2 (12), 3–16.

V. I. Arnautov, Semitopological isomorphisms of topological groups, Bul. Acad. S¸tiint¸e Repub. Mold. Mat. 2004 (2004), no. 1, 15–25.

S. Banach, Ueber metrische Gruppen, Studia Math. 3 (1931), 101–113.

L. Brown, Topologically complete groups, Proc. Amer. Math. Soc. 35 (1972), 593–600.

D. Dikranjan, Recent advances in minimal topological groups, Topology Appl. 85 (1998), no. 1–3, 53–91.

D. Dikranjan and A. Giordano Bruno, Semitopological isomomorphisms for generalized Heisenberg groups, work in progress.

D. Dikranjan and M. Megrelishvili, Relative minimality and co-minimality of subgroups in topological groups, Topology Appl., to appear.

D. Dikranjan, I. Prodanov and L. Stoyanov, Topological Groups: Characters, Dualities and Minimal Group Topologies, Pure and Applied Mathematics, Vol. 130, Marcel Dekker Inc., New York-Basel, 1989.

D. Dikranjan and D. Shakhmatov, Selected topics from the structure theory of topological groups, in: E. Perl, Open Problems in Topology 2, Elsevier (2007), 389–406.

D. Dikranjan and V. Uspenskij, Categorically compact topological groups, J. Pure Appl. Algebra 126 (1998), no. 1–3, 149–168.

D. Doıtchinov, Produits de groupes topologiques minimaux, Bull. Sci. Math. 97 (1972), no. 2, 59–64.

R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.

L. Fuchs, Infinite abelian groups, vol. I, Academic Press New York and London, 1973.

A. Giordano Bruno, Semitopological homomorphisms, Rend. Semin. Mat. Univ. Padova 120 (2008), 79–126.

D. L. Grant, Topological groups which satisfy an open mapping theorem, Pacific J. Math. 68 (1977), 411–423.

T. Husain, Introduction to topological groups, Saunders, Philadelphia, 1966.

H. Kowalski, Beitrage sur topologischen albegra, Math. Naschr. 11 (1954), 143–185.

G. Luk´acs, Hereditarily non-topologizable groups, arXiv:math/0603513v1 [math.GR].

M. Megrelishvili, Generalized Heisenberg groups and Shtern’s question, Georgian Math. J. 11 (2004), no. 4, 775–782.

H. Neumann, Varieties of groups, Springer-Verlag New York, Inc., New York, 1967, x+192 pp.

A. Yu. Ol′shanskii, A remark on a countable non-topologized group, Vestnik Moskov Univ. Ser. I Mat. Mekh. (1980), 103 (in Russian).

V. Pt´ak, Completeness and the open mapping theorem, Bull. Soc. Math. France 86 (1958), 41–74.

D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag, Berlin, 1982.

S. Shelah, On a problem of Kurosh, Jonsson groups and applications, Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), pp. 373ˆu-394, Stud. Logic Foundations Math., 95, North-Holland, Amsterdam-New York (1980).

M. Shlossberg, Minimality on Topological Groups and Heisenberg Type Groups, submitted.

R. M. Stephenson, Jr., Minimal topological groups, Math. Ann. 192 (1971), 193–195.

L. Sulley, A note on B- and Br-complete topological Abelian groups, Proc. Cambr. Phil. Soc. 66 (1969), 275–279.

A. D. Ta˘ımanov, Topologizable groups. II. (Russian) Sibirsk. Mat. Zh. 19 (1978), no. 5, 1201ˆu-1203, 1216. (English translation: Siberian Math. J. 19 no. 5 (1978), 848–850 (1979).)

M. G. Tkachenko, Completeness of topological groups (Russian), Sibirsk. Mat. Zh. 25 (1984), no. 1, 146–158.

M. G. Tkachenko, Some properties of free topological groups (Russian), Mat. Zametki 37 (1985), no. 1, 110–118, 139.

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