A tiny peculiar Fréchet space. (English) Zbl 0543.54021

The authors construct an example of a countable sequentially regular Fréchet space which fails to be regular (X is called sequentially regular) if for each \(f\in C(X) x_ n\to x\Leftrightarrow f(x_ n)\to f(x)\).
Reviewer: A.I.Bashkirov


54D55 Sequential spaces
54G20 Counterexamples in general topology
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
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[1] R. Engelking: General topology. Warszawa 1977. · Zbl 0373.54002
[2] S. P. Franklin: Spaces in which sequences suffice. Fund. Math. 57 (1965), 107-115. · Zbl 0132.17802
[3] M. Fréchet: Sur quelques points du calcul functionel. Rend. Cire. Mat. Palermo 22 (1906), 1-74. · JFM 37.0348.02
[4] R. Frič: A note on Fréchet spaces. Comment. Math. Univ. Carolinae 13 (1972), 411 to 418.
[5] R. Frič: Further note on Fréchet spaces. Comment. Math. Univ. Carolinae 14 (1973), 661-667. · Zbl 0267.54024
[6] R. Frič: On E-sequentially regular spaces. Czechoslovak Math. J. 26 (101) (1976), 604-612. · Zbl 0339.54005
[7] F. Frič V. Koutník: Sequential structures. Convergence structures and applications to analysis. Abh. Akad. Wiss. DDR, Abt. Math. - Naturwiss. - Technik, 1979, N4 4N. Akademie Verlag, Berlin 1980, 37-56.
[8] F. Frič V. Koutník: Sequential convergence since Kanpur conference. General Topology and its Relations to Modern Analysis and Algebra. V (Proc. Fifth Prague Topological Sympos., Prague, 1981). Heldermann Verlag, Berlin, 1982, 193-205.
[9] R. Frič P. Vojtáš: Variants of complete regularity. Abstracts Amer. Math. Soc. Vol. 2, Number 6, October 1981, # 81T-54-572, 556.
[10] F. B. Jones: Moore spaces and uniform spaces. Proc. Amer. Math. Soc. 9 (1958), 483-486. · Zbl 0091.36102
[11] P. Mikusinski: Problems posed at the conference. Proc. Conf. on Convergence, Szczyrk, 1979, Katowice, 1980, 110-112.
[12] V. Koutník: On sequentially regular convergence spaces. Czechoslovak Math. J. 17 [92] (1967), 232-247. · Zbl 0173.24903
[13] J. Novák: Regulární prostor, na němž je každá spojita funkce konstantní. Čas. pěst. mat. a fyz. 75 (1948), 58-68.
[14] J. Novák: On convergence spaces and their sequential envelopes. Czechoslovak Math. J. 15 (90) (1965), 74-100. · Zbl 0139.15906
[15] J. Novak: On some problems concerning the convergence spaces and groups. General Topology and its Relations to Modern Analyzis and Algebra (Proc. Kanpur Topological Conf., 1968). Academia, Praha 1970, 219-229.
[16] F. Siwiec: Generalizations of the first axiom of countability. Rocky Mountain J. Math. 5 (1975), 1-60. · Zbl 0294.54021
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