Weakly metrizable pseudocompact groups

• Dikranjan, Dikran ^[1] ; Giordano Bruno, Anna ^[1] ; Milan, Chiara ^[1]
  1. [1] Università di Udine
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 7, Nº. 1, 2006, págs. 1-39
• Idioma: inglés
• DOI: 10.4995/agt.2006.1930
• Enlaces
  • Texto completo
• Resumen

  • We study various weaker versions of metrizability for pseudocompact abelian groups G: singularity (G possesses a compact metrizable subgroup of the form mG, m > 0), almost connectedness (G is metrizable modulo the connected component) and various versions of extremality in the sense of Comfort and co-authors (s-extremal, if G has no proper dense pseudocompact subgroups, r-extremal, if G admits no proper pseudocompact refinement). We introduce also weaklyextremal pseudocompact groups (weakening simultaneously s-extremal and r-extremal). It turns out that this “symmetric” version of ex-tremality has nice properties that restore the symmetry, to a certain extent, in the theory of extremal pseudocompact groups obtaining simpler uniform proofs of most of the known results. We characterize doubly extremal pseudocompact groups within the class of s-extremal pseudocompact groups. We give also a criterion for r-extremality for connected pseudocompact groups.

• Referencias bibliográficas
  • W.W. Comfort, Tampering With Pseudocompact Groups, Plenary talk at the 2003 Summer Conference on General Topology and Its Applications (Howard...
  • W. W. Comfort and J. Galindo, Pseudocompact topological group refinements of maximal weight, Proc. Amer. Math. Soc. 131 (2003), 1311–1320....
  • W. W. Comfort and J. Galindo, Extremal pseudocompact topological groups, J. Pure Appl. Algebra 197 (2005) (1-3), 59–81.
  • W. W. Comfort, H. Gladdines and J. van Mill, Proper pseudocompact subgroups of pseudocompact Abelian groups, In: Papers on General Topology...
  • W. W. Comfort, K. H. Hofmann and D. Remus, Topological groups and semigroups, Recent progress in general topology (Prague, 1991), 57–144,...
  • W. W. Comfort and J. van Mill, Concerning connected, pseudocompact Abelian groups, Topology Appl. 33 (1989), 21–45. http://dx.doi.org/10.1016/0166-8641(89)90086-2
  • W. W. Comfort and J. van Mill, Some topological groups with, and some without, proper dense subgroups, Topology Appl. 41 (1991), 3–15. http://dx.doi.org/10.1016/0166-8641(91)90096-5
  • W. W. Comfort and J. van Mill, Extremal pseudocompact abelian groups are compact metrizable, Preprint, 2005.
  • W. W. Comfort and L. C. Robertson, Proper pseudocompact extensions of compact Abelian group topologies, Proc. Amer. Math. Soc 86 (1982), 173–178....
  • W. W. Comfort and L. C. Robertson, Cardinality constraints for pseudocompact and for totally dense subgroups of compact Abelian groups, Pacific...
  • W. W. Comfort and L. C. Robertson, Extremal phenomena in certain classes of totally bounded groups, Dissertationes Math. 272 (1988), 48 pages....
  • W.W. Comfort and K. A. Ross, Topologies induced by groups of characters, Fundamenta Math. 55 (1964), 283–291.
  • W. W. Comfort and K. A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483–496. http://dx.doi.org/10.2140/pjm.1966.16.483
  • W. W. Comfort and V. Saks, Countably compact groups and finest totally bounded topologies, Pacific J. Math. 49 (1973), 33–44. http://dx.doi.org/10.2140/pjm.1973.49.33
  • W. W. Comfort and T. Soundararajan, Pseudocompact group topologies and totally dense subgroups, Pacific J. Math. 100 (1982), 61–84. http://dx.doi.org/10.2140/pjm.1982.100.61
  • D. Dikranjan, Dimension and connectedness in pseudo-compact groups, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) (4), 309–314.
  • D. Dikranjan, Zero-dimensionality of some pseudocompact groups, Proc. Amer. Math. Soc. 120 (1994) (4) 1299–1308.
  • D. Dikranjan and A. Giordano Bruno, Pseudocompact totally dense subgroups, Workshop on Topological Groups, Pamplona Spain (August 2005).
  • D. Dikranjan, I. Prodanov and L. Stojanov, Topological groups (Characters, Dualities, and Minimal group Topologies), Marcel Dekker, Inc.,...
  • D. Dikranjan and D. Shakhmatov, Compact-like totally dense subgroups of compact groups, Proc. Amer. Math. Soc. 114 (1992) (4) 1119–1129.
  • D. Dikranjan and D. Shakhmatov, Algebraic structure of pseudocompact groups, Memoirs Amer. Math. Soc. 133 (1998), 83 pages.
  • J. Galindo, The existence of dense pseudocompact subgroups and of pseudocompact refinements, plenary talk at IV Convegno Italo–Spagnolo di...
  • J. Galindo, Dense pseudocompact subgroups and finer pseudocompact group topologies, Scientiae Math. Japonicae 55 (2001), 627–640.
  • A. Giordano Bruno, Gruppi pseudocompatti estremali, M.Sc. Thesis, Università di Udine (March 2004).
  • E. Hewitt, Rings of real-valued continuous functions I, Trans. Amer. Math. Soc. 64 (1948), 45–99. http://dx.doi.org/10.1090/S0002-9947-1948-0026239-9
  • E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, volume I, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen,...
  • E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, volume I, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen,...
  • V. Kuz’minov, On a hypothesis of P.S. Alexandrov in the theory of topological groups, (in Russian) Doklady Akad. Nauk SSSR 125 (1959), 727–729.
  • A. E. Merzon, A certain property of topological-algebraic categories, (Russian) Uspehi Mat. Nauk 27 (1972), no. 4 (166), 217.
  • M. Tkachenko and I. Yaschenko, Independent group topologies on abelian groups, Proceedings of the International Conference on Topology and...
  • E. van Douwen, The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality, Proc. Amer. Math. Soc. 80 (1980),...
  • A.Weil, Sur les Espaces à Structure Uniforme et sur la Topologie Générale, Publ. Math. Univ. Strasbourg, vol. 551, Hermann & Cie, Paris...
  • H. J. Wilcox, Pseudocompact groups, Pacific J. Math. 19 (1966), 365–379. http://dx.doi.org/10.2140/pjm.1966.19.365