## Regularity and extension of maps.(English)Zbl 0762.54019

Regular topological spaces $$Y$$ are known to be characterized by the property that a mapping $$f:X\to Y$$ is continuous provided, for a given dense set $$X_ 0$$ of $$X$$, its restriction to every subspace $$X_ 0\cup \{x\}$$ is continuous. The authors show that this characterization is also true for convergence spaces if one uses strict dense sets $$X_ 0$$ only. This description is then used to define regularity in $$\mathcal L$$-spaces; an inner characterization of such regular $$\mathcal L$$-spaces is given.
Reviewer: M.Hušek (Praha)

### MSC:

 54D10 Lower separation axioms ($$T_0$$–$$T_3$$, etc.) 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 54C20 Extension of maps

### Keywords:

regular space; convergence space
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### References:

 [1] BEATTIE R., BUTZMANN H.-P.: Sequentially determined convergence spaces. Czechoslovak Math. J. 37 (1987), 231-247. · Zbl 0652.54001 [2] BOURBAKI, N: Elements of Mathematics. General Topology. Part I., Addison-Wesley Publ. Co., Reading, 1966. · Zbl 0301.54002 [3] ČECH E.: Topological Spaces. Academia, Praha, 1966. · Zbl 0141.39401 [4] FRIČ R.: Regularity and extension of mappings in sequential spaces. Comment. Math. Univ. Carolin. 15 (1974), 161-171. · Zbl 0275.54019 [5] FRIČ R., KENT D. C.: A completion functor for Cauchy groups. Internat. J. Math. Math. Sci. 4 (1981), 55-65. · Zbl 0465.54002 [6] FRIČ R., KENT D. C.: A nice completion need not be strict. General Topology and its Relations to Modern Analysis and Algebra, V. (Prague, 1981), Sigma Ser. Pure Math. 3, Heldermann, Berlin, 1983, pp. 189-192. · Zbl 0501.54023 [7] FRIČ R., KENT D. C.: Regular L-spaces. · Zbl 0797.54005 [8] FRIČ R., NOVÁK J.: A tiny peculiar Fréchet space. Czechoslovak Math. J. 34 (1984), 22-27. · Zbl 0543.54021 [9] FRIČ R., VOJTÁŠ P.: Diagonal conditions in sequential convergence. Convergence Structures 1984 (Proc. Conf. on Convergence, Bechyne, 1984). Mathematical research/Mathematische Forschung Bd. 24, Akademie-Verlag, Berlin, 1985, pp. 77-94. [10] FRIČ R., ZANOLIN F.: Strict completions of $$L_0^\ast$$-groups. · Zbl 0797.54007 [11] KENT D. C., RICHARDSON G. D.: Regular completions of Cauchy spaces. Pacific J. Math. 51 (1974), 483-490. · Zbl 0291.54024 [12] KRATOCHVÍL P.: Multisequences and their structure is sequential spaces. Convergence Structures 1984 (Proc. Conf. on Convergence, Bechyně, 1984). Mathematical Research/Mathematische Forschung Bd. 24, Akademie-Verlag, Berlin, 1985, pp. 205-216. [13] NOVÁK J.: Regular space on which every continuous function is constant. (Czech), Časopis Pěst. Mat. Fys. 73 (1948), 58-68. · Zbl 0032.43103 [14] NOVÁK J.: Über die eindeutigen stetigen Erweiterungen stetiger Funktionen. Czechoslovak Math. J. 8 (1958), 344-355. · Zbl 0087.37501 [15] NOVÁK J.: Convergence of double sequences. Convergence Structures 1984. (Proc. Conf. on Convergence, Bechyně, 1984), Mathematical Research/Mathematische Forschung Bd. 24, Akademie-Verlag, Berlin, 1985, pp. 233-243. [16] POCHCIAL J.: An example of convergence linear space. Generalized Functions and Convergence. (Proc. Conference, Katowice, 1988), World Scientific, Singapore, 1990, pp. 361-364. [17] PREUSS G.: Theory of Topological Structures. D. Reidel Publ. Co., Dordrecht, 1987.
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