| 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:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
|
[PAU] Paulík, L.: Strictness of $L_0$-ring completions.Tatra Mountains Math. Publ. 5 (1995), 169–175. MR 1384806 |
| . |
