Weakly metrizable pseudocompact groups

Dikran Dikranjan, Anna Giordano Bruno, Chiara Milan


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.


Pseudocompact group; Gδ-dense subgroup; Extremal pseudocompact group; Dense graph

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W.W. Comfort, Tampering With Pseudocompact Groups, Plenary talk at the 2003 Summer Conference on General Topology and Its Applications (Howard University, Washington, DC), Topology Proc. 28 (2) (2004) 401–424.

W. W. Comfort and J. Galindo, Pseudocompact topological group refinements of maximal weight, Proc. Amer. Math. Soc. 131 (2003), 1311–1320. http://dx.doi.org/10.1090/S0002-9939-02-06650-9

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 and Applications, Annals of the New York Academy of Sciences 728 (New York) (Susan Andima, Gerald Itzkowitz, T. Yung Kong, Ralph Kopperman, Prabud Ram Misra, Lawrence Narici, and Aaron. Todd, eds.), pp. 237–247, 1994. [Proc. June, 1992 Queens College Summer Conference on General Topology and Applications.]

W. W. Comfort, K. H. Hofmann and D. Remus, Topological groups and semigroups, Recent progress in general topology (Prague, 1991), 57–144, North-Holland, Amsterdam, 1992.

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. http://dx.doi.org/10.1090/S0002-9939-1982-0663891-4

W. W. Comfort and L. C. Robertson, Cardinality constraints for pseudocompact and for totally dense subgroups of compact Abelian groups, Pacific J. Math. 119 (1985), 265–285. http://dx.doi.org/10.2140/pjm.1985.119.265

W. W. Comfort and L. C. Robertson, Extremal phenomena in certain classes of totally bounded groups, Dissertationes Math. 272 (1988), 48 pages. [Rozprawy Mat. Polish Scientific Publishers, Warszawa.]

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., New York-Basel (1990).

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 Topologia Generale e le sue Applicazioni, Bressanone (June 26–30, 2001).

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, volume 115, Springer Verlag, Berlin-G¨ottingen-Heidelberg (1963).

E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, volume I, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 152, Springer-Verlag, Berlin-Heidelberg-New York (1970).

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 its Applications (Yokohama, 1999). Topology Appl. 122 (2002) (1-2), 425–451.

E. van Douwen, The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality, Proc. Amer. Math. Soc. 80 (1980), 678–682. http://dx.doi.org/10.1090/S0002-9939-1980-0587954-5

A.Weil, Sur les Espaces à Structure Uniforme et sur la Topologie Générale, Publ. Math. Univ. Strasbourg, vol. 551, Hermann & Cie, Paris (1938).

H. J. Wilcox, Pseudocompact groups, Pacific J. Math. 19 (1966), 365–379. http://dx.doi.org/10.2140/pjm.1966.19.365

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1. Dense minimal pseudocompact subgroups of compact Abelian groups
Anna Giordano Bruno
Topology and its Applications  vol: 155  issue: 17-18  first page: 1919  year: 2008  
doi: 10.1016/j.topol.2007.04.020

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Universitat Politècnica de València

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