Examples of sequential topological groups under the continuum hypothesis

• Autorzy: Alexander Shibakov
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

topological group; sequential space; Fréchet space

Kategorie tematyczne

• 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)
• 54D55: Sequential spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1996
• Tom: 151
• Numer: 2
• Strony: 107-120

Daty

wydano

1996

otrzymano

1995-05-03

poprawiono

1996-05-31

Twórcy

autor

Alexander Shibakov

• Department of Mathematics, Auburn University, Auburn, Alabama 36849, U.S.A.

Bibliografia

• [A1] A. Arkhangel'skiĭ, The frequency spectrum of a topological space and the classification of spaces, Soviet Math. Dokl. 13 (1972), 265-268.
• [A2] A. Arkhangel'skiĭ, Topological properties in topological groups, in: XVIII All Union Algebraic Conference, Kishinev, 1985 (in Russian).
• [A3] A. Arkhangel'skiĭ, The frequency spectrum of a topological space and the product operation, Trans. Moscow Math. Soc. 2 (1981), 163-200.
• [AF] A. Arkhangel'skiĭ and S. Franklin, Ordinal invariants for topological spaces, Michigan Math. J. 15 (1968), 313-320.
• [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.
• [BM] D. Burke and E. Michael, On a theorem of V. V. Filippov, Israel J. Math. 11 (1972), 394-397.
• [vD] E. K. van Douwen, The product of a Fréchet space and a metrizable space, Topology Appl. 47 (1992), 163-164.
• [DS] A. Dow and J. Steprāns, Countable Fréchet $α_1$-spaces may be first-countable, Arch. Math. Logic 32 (1992), 33-50.
• [EKN] K. Eda, S. Kamo and T. Nogura, Spaces which contain a copy of the rationals, J. Math. Soc. Japan 42 (1990), 103-112.
• [F] S. Franklin, Spaces in which sequences suffice, Fund. Math. 57 (1965), 107-115.
• [GMT] G. Gruenhage, E. Michael and Y. Tanaka, Spaces determined by point-countable covers, Pacific J. Math. 113 (1984), 303-332.
• [MS] V. Malykhin and B. Shapirovskiĭ, Martin's axiom and properties of topological spaces, Soviet Math. Dokl. 14 (1973), 1746-1751.
• [M1] E. Michael, $ℵ_0$-spaces, J. Math. Mech. 15 (1966), 983-1002.
• [M2] E. Michael, A quintuple quotient quest, Gen. Topology Appl. 2 (1972), 91-138.
• [No1] T. Nogura, The product of $⟨α_i⟩$-spaces, Topology Appl. 21 (1985), 251-259.
• [No2] T. Nogura, Products of sequential convergence properties, Czechoslovak Math. J. 39 (1989), 262-279.
• [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.
• [NST2] T. Nogura, D. Shakhmatov and Y. Tanaka, $α_4$-property versus A-property in topological spaces and groups, to appear.
• [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.
• [N] P. J. Nyikos, Metrizability and Fréchet-Urysohn property in topological groups, Proc. Amer. Math. Soc. 83 (1981), 793-801.
• [O] R. C. Olson, Bi-quotient maps, countably bi-sequential spaces, and related topics, Gen. Topology Appl. 4 (1974), 1-28.
• [R] M. Rajagopalan, Sequential order and spaces $S_n$, Proc. Amer. Math. Soc. 54 (1976), 433-438.
• [Sm] D. Shakhmatov, $α_i$-properties in Fréchet-Urysohn topological groups, Topology Proc. 15 (1990), 143-183.
• [Sh] A. Shibakov, A sequential group topology on rationals with intermediate sequential order, Proc. Amer. Math. Soc. 124 (1996), 2599-2607.
• [Si] P. Simon, A compact Fréchet space whose square is not Fréchet, Comment. Math. Univ. Carolin. 21 (1980), 749-753.
• [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.