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Sequential completeness of quotient groups

Published online by Cambridge University Press:  17 April 2009

Dikran Dikranjan  and
Michael Tkačenko
Dikran Dikranjan
Affiliation:
Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy, e-mail: [email protected]
Michael Tkačenko
Affiliation:
Departamento de Matemáticas, Universidad Autónoma Metropolitana, México, e-mail: [email protected]
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Abstract

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We discuss various generalisations of countable compactness for topological groups that are related to completeness. The sequentially complete groups form a class closed with respect to taking direct products and closed subgroups. Surprisingly, the stronger version of sequential completeness called sequential h-completeness (all continuous homomorphic images are sequentially complete) implies pseudocompactness in the presence of good algebraic properties such as nilpotency. We also study quotients of sequentially complete groups and find several classes of sequentially q-complete groups (all quotients are sequentially complete). Finally, we show that the pseudocompact sequentially complete groups are far from being sequentially q-complete in the following sense: every pseudocompact Abelian group is a quotient of a pseudocompact Abelian sequentially complete group.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 61 , Issue 1 , February 2000 , pp. 129 - 150
Copyright
Copyright © Australian Mathematical Society 2000

References

REFERENCES

[1]Arhangel'skiǏ, A.V., ‘Cardinal invariants of topologial groups. Embeddings and condensations’, Soviet Math. Dokl. 20 (1979), 783–787. Russian original in: Dokl. AN SSR 247 (1979), 779–782.Google Scholar
[2]Arhangel'skiǏ, A.V., ‘Classes of topological groups’, Russian Math. Surveys 36 (1981), 151–174. Russian original in: Uspekhy Mat. Nauk 36 (1981), 127–146.CrossRefGoogle Scholar
[3]Arhangel'skiǏ, A.V., ‘Spaces of functions with the pointwise convergence topology and compacta’, (in Russian), Uspekhy Mat. Nauk 39 (1984), 11–50.Google Scholar
[4]Arhangel'skiǏ, A.V. and Ponomarev, V.I., ‘On dyadic compacta’, (in Russian), Dokl. Akad. Nauk SSSR 182 (1968), 993–996.Google Scholar
[5]Comfort, W.W., ‘Problems on topological groups and other homogeneous spaces’, in Open problems in topology, (van Mill, J. and Reed, G. M., Editors) (North-Holland, Amsterdam, 1990), pp. 313–347.Google Scholar
[6]Comfort, W.W. and van Mill, J., ‘On the existence of free topological groups’, Topology Appl. 29 (1988), 245–269.CrossRefGoogle Scholar
[7]Comfort, W.W., Morris, S.A., Robbie, D., Svetlichny, S. and Tkačenko, M.G., ‘Suitable sets for topological groups’, Topology Appl. 86 (1998), 25–46.CrossRefGoogle Scholar
[8]Comfort, W.W. and Robertson, L., ‘External phenomena in certain classes of totally bounded groups’, Dissert. Math. 272 (1988), 1–48.Google Scholar
[9]Comfort, W.W. and Ross, K., ‘Pseudocompactness and uniform continuity in topological groups’, Pacific J. Math. 16 (1966), 483–496.CrossRefGoogle Scholar
[10]Comfort, W.W. and Saks, V., ‘Countably compact groups and the finest totally bounded topologies’, Pacific J. Math. 49, (1973), 33–44.CrossRefGoogle Scholar
[11]Comfort, W.W. and Trigos-Arrieta, F.J., ‘Remarks on a theorem of Glicksberg’, in General topology and its applications, (Andima, S.J., Kopperman, R., Misra, P.R., Reichman, J.Z. and Todd, A.R., Editors), Lecture Notes in Pure and Applied Mathematics 134 (Marcel Dekker, New York, 1991), pp. 25–33.Google Scholar
[12]Corson, H.H., ‘Normality of subsets of product spaces’, Amer. J. Math. 81 (1959), 785–796.CrossRefGoogle Scholar
[13]Dikranjan, D., ‘Zero-dimensionality of some pseudocompact groups’, Proc. Amer. Math. Soc. 120 (1994), 1299–1308.CrossRefGoogle Scholar
[14]Dikranjan, D., ‘Countably compact groups satisfying the open mapping theorem’, Topology Appl. (to appear).Google Scholar
[15]Dikranjan, D., Prodanov, Iv. and Stoyanov, L., Topological groups: Character, dualities and minimal group topologies, Pure and Applied Mathematics 130 (Marcel Dekker Inc., New York, Basel, 1989).Google Scholar
[16]Dikranjan, D. and Tholen, W., Categorical structure of closure operators with applications to topology, Algebra and discrete mathematics, Mathematics and its Applications 346 (Kluwer Academic Publishers, Dordrecht, Boston, London, 1995).CrossRefGoogle Scholar
[17]Dikranjan, D. and Tkačenko, M., ‘Sequently complete groups: dimension and minimality’, J. Pure. Appl. Algebra (to appear).Google Scholar
[18]Dikranjan, D. and Tkačenko, M., ‘Weak completeness of free topological groups’, Topology Appl. (to appear).Google Scholar
[19]Dikranjan, D. and Tonolo, A., ‘On a characterization of linear compactness’, Riv. Mat. Pura Appl. 17 (1995), 95–106.Google Scholar
[20]Dikranjan, D. and Uspenskiǐ, V., ‘Categorically compact topological groups’, J. Pure Appl. Algebra 85 (1998), 53–91.Google Scholar
[21]van Douwen, E. K., ‘The product of two countably compact topological groups’, Trans. Amer. Math. Soc. 262 (1980), 417–427.CrossRefGoogle Scholar
[22]van Douwen, E. K., ‘The maximal totally bounded group topology on G and the biggest minimal G’, Topology Appl. 34 (1990), 69–91.CrossRefGoogle Scholar
[23]Flor, P., ‘Zur Bohr-Konvergenz von Folgen’, Math. Scand. 23 (1968), 169–170.CrossRefGoogle Scholar
[24]Franklin, S.P. and Tomas, B.V. Smith, ‘A survey of k ω-spaces’, Topology Proc. 2 (1977), 111–124.Google Scholar
[25]Glicksberg, I., ‘Uniform boundedness for groups’, Canad. J. Math. 14 (1962), 269–276.CrossRefGoogle Scholar
[26]Graev, M.I., ‘Free topological groups’, Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 279–323. English translation in: Topology and Topological Algebra, Translation Series 1, 8 American Mathematical Society (1962), pp. 305–364.Google Scholar
[27]Grant, D.L., ‘Topological groups which satisfy an open mapping theorem’, Pacific J. Math. 68 (1977), 411–423.CrossRefGoogle Scholar
[28]Gul'ko, S.P., ‘On properties of subsets of Σ-products’, (in Russian), Dokl. Akad. Nauk. SSSR 237 (1977).Google Scholar
[29]Guran, I., ‘On topological groups close to being Lindelöf’, Soviet Math. Dokl. 23 (1981), 173–175.Google Scholar
[30]Hofmann, K.-H. and Morris, S.A., ‘Weight and c’, J. Pure Appl. Algebra 68 (1990), 181–194.CrossRefGoogle Scholar
[31]Hunt, D.C. and Morris, S.A., ‘Free subgroups of free topological groups’,in Proc. Second Internat. Conf. on the Theory of Groups,Canberra,1973, Lecture Notes in Mathematics 372 (Springer-Verlag, Berlin, Heidelberg, New York, 1974), pp. 377–387.CrossRefGoogle Scholar
[32]Mack, J., Morris, S.A. and Ordman, E.T., ‘Free topological groups and the projective dimensions of locally compact Abelian group’, Proc. Amer. Math. Soc. 40 (1973), 303–308.CrossRefGoogle Scholar
[33]Morris, S.A., ‘Varieties of topological groups: a survey’, Colloq. Math. 46 (1982), 147–165.CrossRefGoogle Scholar
[34]Noble, N., ‘The continuity of functions on Cartesian products’, Trans. Amer. Math. Soc. 149 (1970), 187–198.CrossRefGoogle Scholar
[35]Nummela, E.C., ‘Uniform free topological groups and Samuel compactifications’, Topology Appl. 13 (1982), 77–83.CrossRefGoogle Scholar
[36]Pestov, V.G., ‘Some properties of free topological groups’, Moscow Univ. Math. Bull. 37 (1982), 46–49.Google Scholar
[37]Prodanov, Iv., ‘Some minimal group topologies are precompact’, Math. Ann. 227 (1977), 117–125.CrossRefGoogle Scholar
[38]Protasov, I.V., ‘Minimal varieties of topological groups’, (in Russian), Dokl. Akad. Nauk Ukrain. SSR Ser. A 8 (1988), 14–16.Google Scholar
[39]Raĭkov, D.A., ‘A criterion of completeness of topological linear spaces and topological Abelian groups’, (in Russian), Mat. Zametky 16 (1974), 101–106.Google Scholar
[40]Remus, D. and Stoyanov, L., ‘Complete minimal and totally minimal groups’, Topology Appl. 42 (1991), 57–69.CrossRefGoogle Scholar
[41]Robinson, D.J.F., A course in the theory of groups (Springer-Verlag, Berlin, Heidelberg, New York, 1982).CrossRefGoogle Scholar
[42]Tkačenko, M.G., ‘Compactness type properties in topological groups’, Czech. Math. J. 38 (1988), 324–341.CrossRefGoogle Scholar
[43]Tkačenko, M.G., ‘Strong collectionwise normality and countable compactness in free topological groups’, Siberian Math. J. 28 (1987), 167–177.Google Scholar
[44]Tkačenko, M.G., ‘Pseudocompact topological groups and their properties’, Siberian Math. J. 30 (1989), 154–164.Google Scholar
[45]Tkačenko, M.G., ‘Countably compact and pseudocompact topologies on free Abelian groups’, Soviet Math. (Iz. VUZ) 34 (1990), 79–86.Google Scholar
[46]Vaughan, J., ‘Countably compact and sequentially compact spaces’, in Handbook of set-theoretic topology, (Kunen, K. and Vaughan, J.E., Editors) (North-Holland, Amsterdam, New York, Oxford, 1984), pp. 569–602.CrossRefGoogle Scholar