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1993 | 143 | 2 | 119-136
Tytuł artykułu

The Bohr compactification, modulo a metrizable subgroup

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The authors prove the following result, which generalizes a well-known theorem of I. Glicksberg [G]: If G is a locally compact Abelian group with Bohr compactification bG, and if N is a closed metrizable subgroup of bG, then every A ⊆ G satisfies: A·(N ∩ G) is compact in G if and only if {aN:a ∈ A} is compact in bG/N. Examples are given to show: (a) the asserted equivalence can fail in the absence of the metrizability hypothesis, even when N ∩ G = {1}; (b) the asserted equivalence can hold for suitable G and N with N closed in bG but not metrizable; (c) an Abelian group may admit two topological group topologies U and T, with U totally bounded, T locally compact,U ⊆ T, with U and T sharing the same compact sets, and such that nevertheless U is not the topology inherited from the Bohr compactification of ⟨ G, T⟩. There are applications to topological groups of the form kG for G a totally bounded Abelian group.
Słowa kluczowe
Rocznik
Tom
143
Numer
2
Strony
119-136
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-04-07
poprawiono
1992-12-02
poprawiono
1993-05-14
Twórcy
  • Department of Mathematics, California State University, Bakersfield, California 93311-1099, U.S.A., jtrigos@csbina.csubak.edu
autor
  • Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459, U.S.A.
Bibliografia
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  • [CR] W. W. Comfort and K. A. Ross, Topologies induced by groups of characters, Fund. Math. 55 (1964), 283-291.
  • [CT] W. W. Comfort and F. J. Trigos-Arrieta, Remarks on a Theorem of Glicksberg, in: General Topology and Applications, S. J. Andima, R. Kopperman, P. R. Misra, J. Z. Reichman and A. R. Todd (eds.), Lecture Notes in Pure and Appl. Math. 134, Dekker, New York, 1991, 25-33.
  • [DPS] D. N. Dikranjan, I. R. Prodanov and L. N. Stoyanov, Topological Groups, Lecture Notes in Pure and Appl. Math. 130, Dekker, New York, 1989.
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  • [F] P. Flor, Zur Bohr-Konvergenz von Folgen, Math. Scand. 23 (1968), 169-170.
  • [G] I. Glicksberg, Uniform boundedness for groups, Canad. J. Math. 14 (1962), 269-276.
  • [HR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. I, Grundlehren Math. Wiss. 115, Springer, Berlin, 1963.
  • [Hu] R. Hughes, Compactness in locally compact groups, Bull. Amer. Math. Soc. 79 (1973), 122-123.
  • [KN] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 1974.
  • [K] V. Kuz'minov, On a hypothesis of P. S. Alexandrov in the theory of topological groups, Dokl. Akad. Nauk SSSR 125 (1959), 727-729 (in Russian).
  • [L] W. F. LaMartin, On the foundations of k-group theory, Dissertationes Math. 146 (1977).
  • [Mo] W. Moran, On almost periodic compactifications of locally compact groups, J. London Math. Soc. (2) 3 (1971), 507-512.
  • [Re] G. A. Reid, On sequential convergence in groups, Math. Z. 102 (1967), 227-235.
  • [RT] D. Remus and F. J. Trigos-Arrieta, Abelian groups which satisfy Pontryagin duality need not respect compactness, Proc. Amer. Math. Soc. 117 (1993), 1195-1200.
  • [Ro] K. A. Ross, Commentary on selected papers of S. Kakutani, in: S. Kakutani, Selected Papers, Vol. I, Robert A. Kallman (ed.), Birkhäuser, Boston, 1986.
  • [Š] B. È. Šapirovskiĭ, On embedding extremally disconnected spaces in compact Hausdorff spaces. b-points and weight of pointwise normal spaces, Dokl. Akad. Nauk SSSR 223 (1975), 1083-1086 (in Russian); English transl.: Soviet Math. Dokl. 16 (1975), 1056-1061.
  • [Sc] I. J. Schoenberg, Asymptotic distribution of sequences (solution of Problem 5090), Amer. Math. Monthly 71 (1964), 332-334.
  • [Sh] D. B. Shakhmatov, A direct proof that every infinite compact group G contains ${0,1}^w(G)$, in: Papers in General Topology and Applications, Proc. June 1992 Queens College Conference, Ann. New York Acad. Sci., New York, to appear.
  • [Sm] Yu. M. Smirnov, On the metrizability of bicompacta decomposable into a union of sets with countable basis, Fund. Math. 43 (1956), 387-393 (in Russian).
  • [Sto] A. H. Stone, Metrisability of unions, Proc. Amer. Math. Soc. 10 (1959), 361-366.
  • [Str] K. R. Stromberg, Universally nonmeasurable subgroups of ℝ, Amer. Math. Monthly 99 (1992), 253-255.
  • [T1] F. J. Trigos-Arrieta, Pseudocompactness on groups, doctoral dissertation, Wesleyan University, 1991.
  • [T2] F. J. Trigos-Arrieta, Pseudocompactness on groups, in: General Topology and Applications, Lecture Notes in Pure and Appl. Math. 134, S. Andima et al. (eds.), Dekker, New York, 1991, 369-378.
  • [T3] F. J. Trigos-Arrieta, Continuity, boundedness, connectedness and the Lindelöf property for topological groups, J. Pure Appl. Algebra 70 (1991), 199-210 (= Proceedings of the 1989 Curaçao Conference on Locales and Topological Groups).
  • [Ve] R. Venkataraman, Compactness in Abelian topological groups, Pacific J. Math. 57 (1975), 591-595.
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  • [W] A. Weil, Sur les espaces à structure uniforme et sur la topologie générale, Publ. Math. Univ. Strasbourg, Hermann, Paris, 1937.
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Bibliografia
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bwmeta1.element.bwnjournal-article-fmv143i2p119bwm
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