# Article

 Title: Rings of maps: sequential convergence and completion (English) Author: Frič, Roman Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 49 Issue: 1 Year: 1999 Pages: 111-118 Summary lang: English . Category: math . Summary: The ring $B(R)$ of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring $C(R)$ of all continuous functions and, similarly, the ring $\mathbb{B}$ of all Borel measurable subsets of $R$ is a sequential ring completion of the subring $\mathbb{B}_0$ of all finite unions of half-open intervals; the two completions are not categorical. We study $\mathcal L_0^*$-rings of maps and develop a completion theory covering the two examples. In particular, the $\sigma$-fields of sets form an epireflective subcategory of the category of fields of sets and, for each field of sets $\mathbb{A}$, the generated $\sigma$-field $\sigma (\mathbb{A})$ yields its epireflection. Via zero-rings the theory can be applied to completions of special commutative $\mathcal L_0^*$-groups. (English) Keyword: Rings of sets Keyword: completion of sequential convergence rings Keyword: $Z(2)$-generation Keyword: $Z(2)$-completion Keyword: $\sigma$-rings of maps Keyword: epireflection Keyword: fields of events Keyword: foundation of probability MSC: 54A20 MSC: 54B30 MSC: 54H13 MSC: 60A99 idZBL: Zbl 0949.54003 idMR: MR1676833 . Date available: 2009-09-24T10:20:25Z Last updated: 2020-07-03 Stable URL: http://hdl.handle.net/10338.dmlcz/127471 . Reference: [BKF] Borsík, J. and Frič, R.: Pointwise convergence fails to be strict.Czechoslovak Math. J. 48(123) (1998), 313–320. MR 1624327, 10.1023/A:1022841621251 Reference: [FCA] Frič, R.: On continuous characters of Borel sets.In Proceedings of the Conference on Convergence Spaces (Univ. Nevada, Reno, Nev., 1976), Dept. Math. Univ. Nevada, Reno, Nev., 1976, pp. 35–44. MR 0437673 Reference: [FCB] Frič, R.: On completions of rationals.In Recent Developments of General Topology and its Applications, Math. Research No. 67, Akademie-Verlag, Berlin, 1992, pp. 124–129. MR 1219772 Reference: [FKO] Frič, R. and Koutník, V.: Completions for subcategories of convergence rings.In Categorical Topology and its Relations to Modern Analysis, Algebra and Combinatorics, World Scientific Publishing Co., Singapore, 1989, pp. 195–207. MR 1047901 Reference: [FKT] Frič, R. and Koutník, V.: Sequential convergence spaces: iteration, extension, completion, enlargement.In Recent Progress in General Topology, North Holland, Amsterdam, 1992, pp. 199–213. MR 1229126 Reference: [FMR] Frič, R., McKennon, K. and Richardson, G. D.: Sequential convergence in $C(X)$.In Convergence Structures and Application to Analysis (Frankfurt/Oder, 1978), Abh. Akad. Wiss. DDR, Abt. Math.-Naturwiss.-Technik, 1979, Nr. 4N, Akademie-Verlag, Berlin, 1980, pp. 57–65. MR 0614001 Reference: [FPA] Frič, R. and Piatka, Ľ.: Continuous homomorphisms in set algebras.Práce Štud. Vys. Šk. Doprav. Žiline Sér. Mat.-fyz., 2 (1979), 13–20. (Slovak) MR 0675939 Reference: [FZE] Frič, R. and Zanolin, F.: Coarse sequential convergence in groups, etc..Czechoslovak Math. J. 40 (115) (1990), 459–467. MR 1065025 Reference: [FZS] Frič, R. and Zanolin, F.: Strict completions of $L _0^*$-groups.Czechoslovak Math. J. 42 (117) (1992), 589–598. MR 1182190 Reference: [HES] Herrlich, H. and Strecker, G. E.: Category Theory.2nd edition, Heldermann Verlag, Berlin, 1976. MR 2377903 Reference: [ITH] Isbell, J. R. and Thomas Jr., S.: Mazur’s theorem on sequentially continuous functionals.Proc. Amer. Math. Soc. 14 (1963), 644–647. MR 0151833 Reference: [LAC] Laczkovich, M.: Baire 1 functions.Real Analysis Exchange 9 (1983/84), 15–28. 10.2307/44153506 Reference: [NOE] Novák, J.: Über die eindeutigen stetigen Erweiterungen stetiger Funktionen.Czechoslovak Math. J. 8 (1958), 344–355. MR 0100826 Reference: [NOV] Novák, J.: On completions of convergence commutative groups.In General Topology and its Relations to Modern Analysis and Algebra III (Proc. Third Prague Topological Sympos., 1971), Academia, Praha, 1972, pp. 335–340. MR 0365451 Reference: [PAU] Paulík, L.: Strictness of $L_0$-ring completions.Tatra Mountains Math. Publ. 5 (1995), 169–175. MR 1384806 .

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