On Ponomarev-Systems

Ge, Ying; Shou, Lin

Ge, Ying ; Shou, Lin

Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007), p. 455-467 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

In this paper the relations of mappings and families of subsets are investigated in Ponomarev-systems, and the following results are obtained. (1) $f$ is a sequence-covering (resp. 1-sequence-covering) mapping iff $\mathcal{P}$ is a csf -network (resp. snf -network) of $X$ for a Ponomarev-system $(f,M,X,\mathcal{P})$ ; (2) $f$ is a sequence-covering (resp. 1-sequence-covering) mapping iff every $\mathcal{P}_{n}$ is a cs-cover (resp. wsn-cover) of $X$ for a Ponomarev-system $(f,M,X,\{\mathcal{P}_{n}\})$ . As applications of these results, some relations between sequence-covering mappings and 1-sequence-covering mappings are discussed, and a question posed by S. Lin is answered.

In questo lavoro vengono studiate le relazioni fra mappe e famiglie di sottoinsiemi nei sistemi di Ponomarev, e si ottengono i seguenti risultati. (1) $f$ è una "sequence-covering'" (risp. una "1-sequence-covering") mappa se e solo se $\mathcal{P}$ è una csf rete (risp. una snf rete) di $X$ per un sistema di Ponomarev $(f,M,X,\mathcal{P})$ ; (2) $f$ è una "sequence-covering" (risp. una "1-sequence-covering") mappa se e solo se ogni $\mathcal{P}_{n}$ è un cs ricoprimento (risp. un wsn ricoprimento) di $X$ per un sistema di Ponomarev $(f,M,X,\{\mathcal{P}_{n}\})$ . Come applicazione di questi risultati vengono discusse alcune relazioni fra "sequence-covering" mappe e "1-sequence-covering" mappe, e si fornisce la risposta a una domanda posta da S. Lin.

Publié le : 2007-06-01

MR 2339454 | Zbl 1143.54003

@article{BUMI_2007_8_10B_2_455_0,
     author = {Ying Ge and Lin Shou},
     title = {On Ponomarev-Systems},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {10-A},
     year = {2007},
     pages = {455-467},
     zbl = {1143.54003},
     mrnumber = {2339454},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2007_8_10B_2_455_0}
}

Ge, Ying; Shou, Lin. On Ponomarev-Systems. Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007) pp. 455-467. http://gdmltest.u-ga.fr/item/BUMI_2007_8_10B_2_455_0/

Bibliographie

[1] Ge, Y., Spaces with countable sn-networks, Comment Math. Univ. Carolinae, 45 (2004), 169-176. | MR 2076868 | Zbl 1098.54025

[2] Ge, Y., Mappings in Ponomarev-Systems, Topology Proc., 29 (2005), 141-153. | MR 2182923 | Zbl 1085.54018

[3] Gruenhage, G. - Michael, E. - Tanaka, Y., Spaces determined by point-countable covers, Pacific J. Math., 113 (1984), 303-332. | MR 749538 | Zbl 0561.54016

[4] Guthrie, J. A., A characterization of $\aleph_{0}$ -spaces, General Topology Appl., 1 (1971), 105-110. | MR 288726 | Zbl 0216.19103

[5] Lin, S., A note on the Arens' spaces and sequential fan, Topology Appl., 81 (1997), 185-196. | MR 1485766 | Zbl 0885.54019

[6] Lin, S., Point-Countable Covers and Sequence-Covering Mappings, Beijing: Chinese Science Press, 2002 (in Chinese). | MR 1939779 | Zbl 1004.54001

[7] Lin, S., A note on sequence-covering mappings, Acta Math. Hungar., 107 (2005), 187- 191. | MR 2148582 | Zbl 1081.54025

[8] Lin, S. - Yan, P., Sequence-covering maps of metric spaces, Topology Appl., 109 (2001), 301-314. | MR 1807392 | Zbl 0966.54012

[9] Lin, S. - Yan, P., Notes on cfp-covers, Comment Math. Univ. Carolinae, 44 (2003), 295- 306. | MR 2026164 | Zbl 1100.54021

[10] Michael, E., $\aleph_{0}$ -spaces, J. Math. Mech., 15 (1966), 983-1002. | MR 206907

[11] Ponomarev, V. I., Axiom of countability and continuous mappings, Bull. Pol. Acad. Math., 8 (1960), 127-133. | MR 116314 | Zbl 0095.16301

[12] Siwiec, F., Sequence-covering and countably bi-quotient mappings, General Topology Appl., 1 (1971), 143-154. | MR 288737 | Zbl 0218.54016

[13] Tanaka, Y. - Ge, Y., Around quotient compact images of metric spaces, and symmetric spaces, Houston J. Math., 32 (2006), 99-117. | MR 2202355 | Zbl 1102.54034

[14] Yan, P., On strong sequence-covering compact mapppings, Northeastem Math J., 14 (1998), 341-344. | MR 1685267 | Zbl 0927.54030