Examples of sequential topological groups under the continuum hypothesis

Shibakov, Alexander

Fundamenta Mathematicae, Tome 149 (1996), p. 107-120 / Harvested from The Polish Digital Mathematics Library

Résumé

Using CH we construct examples of sequential topological groups: 1. a pair of countable Fréchet topological groups whose product is sequential but is not Fréchet, 2. a countable Fréchet and $α_{1}$ topological group which contains no copy of the rationals.

Publié le : 1996-01-01

Zbl 0879.54030

EUDML-ID : urn:eudml:doc:212184

@article{bwmeta1.element.bwnjournal-article-fmv151i2p107bwm,
     author = {Alexander Shibakov},
     title = {Examples of sequential topological groups under the continuum hypothesis},
     journal = {Fundamenta Mathematicae},
     volume = {149},
     year = {1996},
     pages = {107-120},
     zbl = {0879.54030},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv151i2p107bwm}
}

Shibakov, Alexander. Examples of sequential topological groups under the continuum hypothesis. Fundamenta Mathematicae, Tome 149 (1996) pp. 107-120. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv151i2p107bwm/

Bibliographie

[00000] [A1] A. Arkhangel'skiĭ, The frequency spectrum of a topological space and the classification of spaces, Soviet Math. Dokl. 13 (1972), 265-268.

[00001] [A2] A. Arkhangel'skiĭ, Topological properties in topological groups, in: XVIII All Union Algebraic Conference, Kishinev, 1985 (in Russian).

[00002] [A3] A. Arkhangel'skiĭ, The frequency spectrum of a topological space and the product operation, Trans. Moscow Math. Soc. 2 (1981), 163-200.

[00003] [AF] A. Arkhangel'skiĭ and S. Franklin, Ordinal invariants for topological spaces, Michigan Math. J. 15 (1968), 313-320. | Zbl 0167.51102

[00004] [BR] T. Boehme and M. Rosenfeld, An example of two compact Fréchet Hausdorff spaces whose product is not Fréchet, J. London Math. Soc. 8 (1974), 339-344. | Zbl 0289.54026

[00005] [BM] D. Burke and E. Michael, On a theorem of V. V. Filippov, Israel J. Math. 11 (1972), 394-397. | Zbl 0236.54015

[00006] [vD] E. K. van Douwen, The product of a Fréchet space and a metrizable space, Topology Appl. 47 (1992), 163-164. | Zbl 0759.54013

[00007] [DS] A. Dow and J. Steprāns, Countable Fréchet $α_{1}$ -spaces may be first-countable, Arch. Math. Logic 32 (1992), 33-50. | Zbl 0798.03052

[00008] [EKN] K. Eda, S. Kamo and T. Nogura, Spaces which contain a copy of the rationals, J. Math. Soc. Japan 42 (1990), 103-112. | Zbl 0709.54012

[00009] [F] S. Franklin, Spaces in which sequences suffice, Fund. Math. 57 (1965), 107-115. | Zbl 0132.17802

[00010] [GMT] G. Gruenhage, E. Michael and Y. Tanaka, Spaces determined by point-countable covers, Pacific J. Math. 113 (1984), 303-332. | Zbl 0561.54016

[00011] [MS] V. Malykhin and B. Shapirovskiĭ, Martin's axiom and properties of topological spaces, Soviet Math. Dokl. 14 (1973), 1746-1751. | Zbl 0294.54006

[00012] [M1] E. Michael, $ℵ_{0}$ -spaces, J. Math. Mech. 15 (1966), 983-1002.

[00013] [M2] E. Michael, A quintuple quotient quest, Gen. Topology Appl. 2 (1972), 91-138. | Zbl 0238.54009

[00014] [No1] T. Nogura, The product of $⟨ α_{i} ⟩$ -spaces, Topology Appl. 21 (1985), 251-259.

[00015] [No2] T. Nogura, Products of sequential convergence properties, Czechoslovak Math. J. 39 (1989), 262-279. | Zbl 0691.54017

[00016] [NST1] T. Nogura, D. Shakhmatov and Y. Tanaka, Metrizability of topological groups having weak topologies with respect to good covers, Topology Appl. 54 (1993), 203-212. | Zbl 0808.54026

[00017] [NST2] T. Nogura, D. Shakhmatov and Y. Tanaka, $α_{4}$ -property versus A-property in topological spaces and groups, to appear. | Zbl 0902.22001

[00018] [NT] T. Nogura and Y. Tanaka, Spaces which contain a copy of $S_{ω}$ or $S_{2}$ and their applications, Topology Appl. 30 (1988), 51-62. | Zbl 0657.54021

[00019] [N] P. J. Nyikos, Metrizability and Fréchet-Urysohn property in topological groups, Proc. Amer. Math. Soc. 83 (1981), 793-801. | Zbl 0474.22001

[00020] [O] R. C. Olson, Bi-quotient maps, countably bi-sequential spaces, and related topics, Gen. Topology Appl. 4 (1974), 1-28. | Zbl 0278.54008

[00021] [R] M. Rajagopalan, Sequential order and spaces $S_{n}$ , Proc. Amer. Math. Soc. 54 (1976), 433-438. | Zbl 0305.54027

[00022] [Sm] D. Shakhmatov, $α_{i}$ -properties in Fréchet-Urysohn topological groups, Topology Proc. 15 (1990), 143-183. | Zbl 0754.54015

[00023] [Sh] A. Shibakov, A sequential group topology on rationals with intermediate sequential order, Proc. Amer. Math. Soc. 124 (1996), 2599-2607. | Zbl 0865.54022

[00024] [Si] P. Simon, A compact Fréchet space whose square is not Fréchet, Comment. Math. Univ. Carolin. 21 (1980), 749-753. | Zbl 0466.54022

[00025] [T] S. Todorčević, Some applications of S- and L-combinatorics, in: The Work of Mary Ellen Rudin, F. D. Tall (ed.), Ann. New York Acad. Sci. 705, 1993, 130-167.