Publication: A class of angelic sequential non-Frechet-Urysohn topological groups
Loading...
Full text at PDC
Publication Date
2007-02-01
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier Science
Citations
Exportar
Abstract
Feechet-Urysohn (briefly F-U) property for topological spaces is known to be highly non-multiplicative: for instance, the square of a compact F-U space is not in general Frechet-Urysohn [P. Simon, A compact Frechet space whose square is not Frechet, Comment. Math. Univ. Carolin. 21 (1980) 749-753. [27]]. Van Douwen proved that the product of a metrizable space by a Frechet-Urysohn space may not be (even) sequential. If the second factor is a topological group this behaviour improves significantly: we have obtained (Theorem 1.6(c)) that the product of a first countable space by a F-U topological group is a F-U space. We draw some important consequences by interacting this fact with Pontryagin duality theory. The main results are the following: (1) If the dual group of a metrizable Abelian group is F-U, then it must be metrizable and locally compact. (2) Leaning on (1) we point out a big class of hemicompact sequential non-Frechet-Urysohn groups, namely: the dual groups of metrizable separable locally quasi-convex non-locally precompact groups. The members of this class are furthermore complete, strictly angelic and locally quasi-convex. (3) Similar results are also obtained in the framework of locally convex spaces. Another class of sequential non-Frechet-Urysohn complete topological Abelian groups very different from ours is given in [E.G. Zelenyuk, I.V. Protasov, Topologies of Abelian groups, Math. USSR Izv. 37 (2) (1991) 445-460. [32]].
Description
UCM subjects
Unesco subjects
Keywords
Citation
A. V. Arhangelskii, Topological Function Spaces, Kluwer Academic, Dordrecht, 1992.
A. V. Arhangelskii, The frequency spectrum of a topological space and the classification of spaces, Dokl. Akad. Nauk SSSR 206 (2) (1972) 1185–1189.
A. V. Arhangelskii, V.I. Ponomarev, On dyadic bicompacta, Soviet Math. Dokl. 9 (1968) 1220–1224.
L. Aussenhofer, Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups, Dissertationes Mathematicae CCCLXXXIV (1999).
V.I. Averbukh, O.G. Smolyanov, The various definitions of the derivative in linear topological spaces, Russian Math. Survey 23 (4) (1968) 67–113.
M. Balanzat, La differentielle d'Hadamard-Fréchet dans les espaces vectoriels topologiques, C. R. Acad. Sci. Paris 251 (1960) 2459–2461.
W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Mathematics, vol. 1466, Springer, Berlin, 1991.
M.J. Chasco, Pontryagin duality for metrizable groups, Arch. Math. 70 (1998) 22–28.
M. Bruguera, E. Martín-Peinador, V. Tarieladze, Eberlein-Smulyan Theorem for Abelian topological groups, J. London Math. Soc. (2) 70 (2004) 341–355.
B. Cascales, J. Kakol, S.A. Saxon, Metrizability vs. Fréchet-Urysohn property, Proc. Amer. Math. Soc. 131 (11) (2003) 3623–3631.
B. Cascales, I. Namioka, The Lindelöf property and σ -fragmentability, Fund. Math. 180 (2003) 161–183.
D.N. Dikranjan, I.R. Prodanov, L.N. Stoyanov, Topological Groups: Characters, Dualities and Minimal Group Topologies, Monographs and Textbooks in Pure and Applied Mathematics, vol. 130, Marcel Dekker, New York, 1990.
D. Dikranjan, E. Martín-Peinador, Around angelic groups, in preparation.
E.K. van Douwen, The product of a Fréchet space and a metrizable space, Topology Appl. 47 (1992) 163–164.
R. Engelking, General Topology, Heldermann, Paris, 1989
S. Hernández, J.M. Mazón, On the sequential spaces of continuous functions, Simon Stevin 60 (4) (1986) 323–328.
E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, I, Springer, Berlin, 1963.
H. Jarchow, Locally Convex Spaces, B.G. Teubner, Stttgart, 1981.
J. Ka̧kol, M. López Pellicer, E. Martín Peinador, V. Tarieladze, Lindelöf spaces C(X) over topological groups, Forum Math., in press.
J. Ka̧kol, S.A. Saxon, The Fréchet-Urysohn property, (LM)-spaces and the strongest locally convex vector topology, Math. Proc. R. Ir. Acad. A 103 (1) (2003) 1–8.
J. Ka̧kol, S.A. Saxon, Montel (DF)-spaces, sequential (LM)-spaces and the strongest locally convex vector topology, J. London Math. Soc. 66 (2) (2002) 388–406.
E. Martín-Peinador, A reflexive admissible topological group must be locally compact, Proc. Amer. Math. Soc. 123 (1995) 3563–3566.
E. Martín-Peinador, V. Tarieladze, A property of Dunford-Pettis type in topological groups, Proc. Amer. Math. Soc. 132 (2004) 1827–1834.
E.A. Michael, A quintuple quotient quest, Gen. Topology Appl. 2 (1972) 91–138.
P.J. Nyikos, Metrizability and the Fréchet-Urysohn property in topological groups, Proc. Amer. Math. Soc. 83 (4) (1981) 793–801.
D. Shakhmatov, Convergence in the presence of algebraic structure, in: M. Hušek, J. van Mill (Eds.), Recent Progress in General Topology II. Elsevier, Amsterdam, 2002, pp. 463–484 (Chapter 17). P. Simon, A compact Fréchet space whose square is not Fréchet, Comment. Math. Univ. Carolin. 21 (1980) 749–753.
M.F. Smith, The Pontrjagin duality theorem in linear spaces, Ann. of Math. 56 (2) (1952) 248–253.
S. Todorcevic, Some applications of S and L combinatorics, in: F.D. Tall (Ed.), The Work of Mary Ellen Rudin, in: Annals of the New York Academy of Sciences, vol. 705, New York Academy of Sciences, New York, 1993, pp. 130–167. S. Todorcevic, C. Uzcategui, Analytic k -spaces, Topology Appl. 146–147 (2005) 511–526.
S. Yamamuro, Differential Calculus in Topological Vector Spaces, Lecture Notes in Math., vol. 374, 1974, pp. 1–179.
E.G. Zelenyuk, I.V. Protasov, Topologies of Abelian groups, Math. USSR Izv. 37 (2) (1991) 445–460.