PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1993 | 142 | 3 | 221-240
Tytuł artykułu

Imposing psendocompact group topologies on Abeliau groups

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, $m(α) ≤ 2^α$. We show:
   Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m$(α)≤ r_0 (G) ≤ γ ≤ 2^α$, or α > ω and $α^ω ≤ r_0(G) ≤ 2^α$, then G admits a pseudocompact group topology of weight α.
 Theorem 4.15. Every connected, pseudocompact Abelian group G with wG = α ≥ ω satisfies $r_0(G) ≥ m(α)$.
 Theorem 5.2(b). If G is divisible Abelian with $2^{r_{0}(G)} ≤ γ$, then G admits at most $2^γ$-many pseudocompact group topologies.
 Theorem 6.2. Let $β = α^ω$ or $β = 2^α$ with β ≥ α, and let $β ≤ γ < κ ≤ 2^β$. Then both $⊕_γℚ$ and the free Abelian group on γ-many generators admit exactly $2^κ$-many pseudocompact group topologies of weight κ. Of these, some $κ^+$-many form a chain and some $2^κ$-many form an anti-chain.
Rocznik
Tom
142
Numer
3
Strony
221-240
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-02-11
poprawiono
1992-09-02
Twórcy
autor
  • Institut Für Mathematik, Universität Hannover, Welfengarten 1, D-3000 Hannover, Germany
Bibliografia
  • [Ban] B. Banaschewski, Local connectedness of extension spaces, Canad. J. Math. 8 (1956), 395-398.
  • [Bau] J. E. Baumgartner, Almost-disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic 10 (1976), 401-439.
  • [BCR] S. Berhanu, W. W. Comfort and J. D. Reid, Counting subgroups and topological group topologies, Pacific J. Math. 116 (1985), 217-241.
  • [CEG] F. S. Cater, P. Erdős and F. Galvin, On the density of λ-box products, General Topology Appl. 9 (1978), 307-312.
  • [C] W. W. Comfort, Topological groups, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan (eds.), North-Holland, Amsterdam 1984, 1143-1263.
  • [CvM] W. W. Comfort and J. van Mill, Concerning connected, pseudocompact Abelian groups, Topology Appl. 33 (1989), 21-45.
  • [CRe1] W. W. Comfort and D. Remus, Long chains of Hausdorff topological group topologies, J. Pure Appl. Algebra 70 (1991), 53-72.
  • [CRe2] W. W. Comfort and D. Remus, Pseudocompact topological group topologies, Abstracts Amer. Math. Soc. 12 (1991), p. 289 [= abstract #91T-54-25].
  • [CRe3] W. W. Comfort and D. Remus, Pseudocompact topological group topologies on Abelian groups, ibid. 12 (1991), p. 321 [= abstract #91T-22-66].
  • [CRob] W. W. Comfort and L. C. Robertson, Cardinality constraints for pseudocompact and for totally dense subgroups of compact topological groups, Pacific J. Math. 119 (1985), 265-285.
  • [CRos1] W. W. Comfort and K. A. Ross, Topologies induced by groups of characters, Fund. Math. 55 (1964), 283-291.
  • [CRos2] W. W. Comfort and K. A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483-496.
  • [DS] D. N. Dikranjan and D. B. Shakhmatov, Pseudocompact topologizations of groups, Zb. Rad. (Niš) 4 (1990), 83-93.
  • [vD] E. K. van Douwen, The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality, Proc. Amer. Math. Soc. 80 (1980), 678-682.
  • [D] R. M. Dudley, Continuity of homomorphisms, Duke Math. J. 28 (1961), 587-594.
  • [Fu] L. Fuchs, Infinite Abelian Groups, Vol. I, Pure Appl. Math. 36, Academic Press, New York 1970.
  • [GJ] L. Gillman and M. Jerison, Rings of Continuous Functions, Graduate Texts in Math. 43, Springer, New York 1976.
  • [Hal] P. R. Halmos, Comment on the real line, Bull. Amer. Math. Soc. 50 (1944), 877-878.
  • [Haw] D. Hawley, Compact group topologies for R, Proc. Amer. Math. Soc. 30 (1971), 566-572.
  • [HI] M. Henriksen and J. R. Isbell, Local connectedness in the Stone-Čech compactification, Illinois J. Math. 1 (1957), 574-582.
  • [HR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. I, Grundlehren Math. Wiss. 115, Springer, Berlin 1963.
  • [J] T. Jech, Set Theory, Academic Press, New York 1978.
  • [M1] M. Magidor, On the singular cardinals problem I, Israel J. Math. 28 (1977), 1-31.
  • [M2] M. Magidor, On the singular cardinals problem II, Ann. of Math. 106 (1977), 517-547.
  • [M] O. Masaveu, doctoral dissertation, Wesleyan University, in preparation.
  • [T1] M. G. Tkachenko, On pseudocompact topological groups, Interim Report of the Prague Topological Symposium 2/1987 (1987), p. 18, Czechoslovak Acad. Sci., Prague 1987.
  • [T2] M. G. Tkachenko, Countably compact and pseudocompact topologies on free Abelian groups, Soviet Math. (Izv. VUZ) 34 (1990), 79-86. Russian original: Izv. Vyssh. Uchebn. Zaved. Mat. 1990 (5) (336), 68-75.
  • [We] A. Weil, Sur les espaces à structure uniforme et sur la topologie générale, Publ. Math. Univ. Strasbourg, Hermann, Paris 1937.
  • [Wu] D. E. Wulbert, A characterization of C(X) for locally connected X, Proc. Amer. Math. Soc. 21 (1969), 269-272.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv142i3p221bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.