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Tytuł książki

Extremal phenomena in certain classes of totally bounded groups

Seria
Rozprawy Matematyczne tom/nr w serii: 272 wydano: 1988
Zawartość
Warianty tytułu
Abstrakty
EN
<br>For various pairs <P,Q> of topological properties such that P ⇒ Q, we consider two questions: (A) Does every topological group topology with P extend properly to a topological group topology with Q, and (B) must a topological group with P have a proper dense subgroup with Q? We obtain negative results and positive results. Principal among the latter is the statement that any pseudocompact group G of uncountable weight which satisfies any of the following three conditions has both a strictly finer pseudocompact topological group topology and a proper dense pseudocompact subgroup: (1) G is O-dimensional and Abelian; (2) $G = H^α$ with α > ω, |H| > 1; (3) G is a dense subgroup of $T^{(ω⁺)}$.
<br>Thwarting our attempts to improve (1), (2) and (3) are examples, for every α > ω, of pseudocompact groups G₀ and G₁ of weight α such that (a) there are surjective φ ∈ Hom(G₀,K) with K compact, φ continuous and open, and a dense, pseudocompact subgroup H of K such that $φ^{-1}(H)$ is not pseudocompact; and (b) G₁ admits no homomorphism onto any non-trivial product.
EN

CONTENTS
0. Introduction..............................................................................................5
1. Notation and results from the literature....................................................6
2. Extending a topology: Some negative results........................................10
3. Finding dense subgroups: Some negative results.................................12
4. Extensions and dense subgroups: Some positive results......................14
5. Extremal pseudocompact Abelian groups: The case $x^p ≡ 1$.............24
6. Recognizing pseudocompact groups.....................................................32
7. Extremal pseudocompact Abelian groups: The 0-dimensional case......37
References................................................................................................41
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 272
Liczba stron
42
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCLXXII
Daty
wydano
1988
Twórcy
  • Department of Mathematics, Wesleyan University, Middletown, CT, 06457 USA
  • Department of Mathematics, Wesleyan University, Middletown, CT, 06457 USA
Bibliografia
  • [1] R. W. Bagley, E. H. Connell, and J. D. McKnight, Jr., On properties characterizing pseudo-compact spaces, Proc. Amer. Math. Soc. 9 (1958), 500-506.
  • [2] Shiferaw Berhanu, W. W. Comfort and J. D. Reid, Counting subgroups and topological group topologies. Pacific J. Math. 16 (1985), 217-241.
  • [3] W. W. Comfort, Topological groups. In: Handbook of General Topology, edited by Kenneth Kunen and Jerry Vaughan, pp. 1143-1263. North-Holland Publ. Co., Amsterdam 1984.
  • [4] W. W. Comfort, Compact groups: subgroups and extensions, In: Proc. (1983) Eger, Hungary, Topological Symposium, Edited by J. Gerlits, North-Holland Publ. Co., Amsterdam, 1986, 183-198.
  • [5] W. W. Comfort, On the "fragmentation" of certain pseudocompact groups, Bull. Greek Math. Soc. 25 (1984), 1-13.
  • [6] W. W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Grundlehren der mathematischen Wissenschaften, vol. 211, Springer-Verlag, Berlin-Heidelberg-New York 1974.
  • [7] W. W. Comfort and Lewis C. Robertson, Proper pseudocompact extensions of compact Abelian group topologies, Proc. Amer. Math. Soc. 86 (1982), 173-178.
  • [8] W. W. Comfort and Lewis C. Robertson, Pseudocompact subgroups and extensions, Abstracts Amer. Math. Soc. 5 (1984), 44. [Abstract 809-22-422.]
  • [9] W. W. Comfort and Lewis C. Robertson, On pseudocompact Abelian groups, Abstracts Amer. Math. Soc. 6 (1985), 41. [Abstract 816-22-206.]
  • [10] W. W. Comfort and Lewis C. Robertson, Cardinality constraints for pseudocompact and for totally dense subgroups of compact Abelian groups. Pacific J. Math. (1985).
  • [11] W. W. Comfort and Lewis C. Robertson, Images and quotients of SO(3,ℝ): remarks on a theorem of van der Waerden, Rocky Mountain J. Math. 17 (1987), 1-13.'
  • [12] W. W. Comfort and Kenneth A. Ross, Topologies induced by groups of characters, Fund. Math. 55 (1964), 283-291.
  • [13] W. W. Comfort and Kenneth A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483-496.
  • [14] W. W. Comfort and Victor Saks, Countably compact groups and finest totally bounded topologies, ibid. 49 (1973), 33-44.
  • [15] W. W. Comfort and T. Soundararajan, Pseudocompact group topologies and totally dense subgroups, ibid. 100 (1982), 61-84.
  • [16] H. H. Corson, Normality in subsets of product spaces, Amer. J. Math. 81 (1959), 785-796.
  • [17] Eric K. van Douwen, Homogeneity of βG if G is a topological group, Colloq. Math. 41 (1979), 193-199.
  • [18] Ryszard Engelking, General Topology, Polska Akademia Nauk, Monografie Matematyczne, vol. 60, Państwowe Wydawnictwo Naukowe-Polish Scientific Publishers, Warszawa 1977.
  • [19] Laszlo Fuchs, Infinite Abelian Groups, vol. 1. Pure and Applied Mathematics, no. 36. Academic Press, New York and London 1970.
  • [20] I. M. Gel'fand and D. Raikov, Irreducible unitary representations of locally bicompact groups (in Russian), Mat. Sb. (N. S.) 13 (55) (1943), 301-316.
  • [21] I. M. Gel'fand and D. Raikov, Irreducible unitary representations of locally bicompact groups (in Russian), Dokl. Akad. Nauk SSSR (N. S.) 42 (1944), 199-201.
  • [22] Leonard Gillman and Meyer Jerison, Rings of Continuous Functions, D. Van Nostrand Co., Inc., Princeton-Toronto-London-New York 1960.
  • [23] Irving Glicksberg, On the representation of functional by integrals, Duke Math. J. 19 (1952), 253-261.
  • [24] Irving Glicksberg, Stone-Čech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369-382.
  • [25] Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis, Vol. I, Grundlehren der math. Wissenschaften volume 115, Springer-Verlag, Berlin-Göttingen-Heidelberg 1963.
  • [26] Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis, Vol. II, Grundlehren der math. Wissenschaften volume 152, Springer-Verlag, New York-Heidelberg-Berlin 1970.
  • [27] Karl Heinrich Hofmann, Finite dimensional submodules of G-modules for a compact group G, Proc. Cambridge Philos. Soc. 65 (1969), 47-52.
  • [28] I. Juhász, Cardinal Functions in Topology - Ten Years Later, Mathematical Centre Tracts 123, Mathematisch Centrum, Amsterdam 1980.
  • [29] J. M. Kister, Uniform continuity and compactness in topological groups, Proc. Amer. Math. Soc. 13 (1962), 37-40.
  • [30] S. Mardesič and P. Papič, Sur les espaces dont tout transformation réelle continue est bornée, Hrvatsko Prirod. Društvo. Glasnik Mat.-Fiz. Astron. Ser. II, 10 (1955), 225-232.
  • [31] J. von Neumann, Almost periodic functions in a group, I, Trans. Amer. Math. Soc. 36 (1934), 445-492.
  • [32] J. von Neumann and E. P. Wigner, Minimally almost periodic groups, Annals of Math. (Series 2) 41 (1940), 746-750.
  • [33] W. Roelcke and S. Dierolf, Uniform Structures on Topological Groups and their Quotients, McGraw-Hill International Book Company, New York 1981.
  • [34] J. de Vries, Pseudocompactness and the Stone-Čech compactification for topological groups, Nieuw Arch. Wisk. (3) 23 (1975), 35-48.
  • [35] B. L. van der Waerden, Stetigkeitssätze für halbeinfache Liesche Gruppen, Math. Z. 36 (1933), 780-786.
  • [36] Andre Weil, Sur les Espaces à Structure Uniforme et sur la Topologie Générale, Publ. Math. Univ. Strasbourg, Hermann & Cie., Paris 1937.
Języki publikacji
EN
Uwagi
Identyfikator YADDA
bwmeta1.element.zamlynska-df3d2472-b3bd-4030-b36b-99e3fd170d38
Identyfikatory
ISBN
83-01-08359-X
ISSN
0012-3862
Kolekcja
DML-PL
Zawartość książki

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