zbMATH — the first resource for mathematics

Rings of maps: sequential convergence and completion. (English) Zbl 0949.54003
Summary: The ring \(B(\mathbb R)\) of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring \(C(\mathbb R)\) of all continuous functions and, similarly, the ring \(\mathbb B\) of all Borel measurable subsets of \(\mathbb R\) 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.

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54B30 Categorical methods in general topology
54H13 Topological fields, rings, etc. (topological aspects)
60A99 Foundations of probability theory
Full Text: DOI EuDML
[1] Borsík, J. and Frič, R.: Pointwise convergence fails to be strict. Czechoslovak Math. J. 48(123) (1998), 313-320. · Zbl 0954.46015
[2] 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.
[3] 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. · Zbl 0797.54006
[4] 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.
[5] 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. · Zbl 0811.54005
[6] 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.
[7] Frič, R. and Piatka, Ľ.: Continuous homomorphisms in set algebras. Práce Štud. Vys. Šk. Doprav. Žiline Sér. Mat.-fyz., 2 (1979), 13-20.
[8] Frič, R. and Zanolin, F.: Coarse sequential convergence in groups, etc. Czechoslovak Math. J. 40 (115) (1990), 459-467. · Zbl 0747.54002
[9] Frič, R. and Zanolin, F.: Strict completions of \(L _0^*\)-groups. Czechoslovak Math. J. 42 (117) (1992), 589-598. · Zbl 0797.54007
[10] Herrlich, H. and Strecker, G. E.: Category Theory. 2nd edition, Heldermann Verlag, Berlin, 1976. · Zbl 1125.18300
[11] Isbell, J. R. and Thomas Jr., S.: Mazur’s theorem on sequentially continuous functionals. Proc. Amer. Math. Soc. 14 (1963), 644-647. · Zbl 0116.31803
[12] Laczkovich, M.: Baire 1 functions. Real Analysis Exchange 9 (1983/84), 15-28. · Zbl 0574.26002
[13] Novák, J.: Über die eindeutigen stetigen Erweiterungen stetiger Funktionen. Czechoslovak Math. J. 8 (1958), 344-355. · Zbl 0087.37501
[14] 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.
[15] Paulík, L.: Strictness of \(L_0\)-ring completions. Tatra Mountains Math. Publ. 5 (1995), 169-175. · Zbl 0858.54036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.