On three equivalences concerning Ponomarev-systems
Ge, Ying
Archivum Mathematicum, Tome 042 (2006), p. 239-246 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

Let $\lbrace {\mathcal P}_n\rbrace $ be a sequence of covers of a space $X$ such that $\lbrace st(x,{\mathcal P}_n)\rbrace $ is a network at $x$ in $X$ for each $x\in X$. For each $n\in \mathbb N$, let ${\mathcal P}_n=\lbrace P_{\beta }:\beta \in \Lambda _n\rbrace $ and $\Lambda _ n$ be endowed the discrete topology. Put $M=\lbrace b=(\beta _n)\in \Pi _{n\in \mathbb N}\Lambda _ n: \lbrace P_{\beta _n}\rbrace $ forms a network at some point $x_b\ in \ X\rbrace $ and $f:M\longrightarrow X$ by choosing $f(b)=x_b$ for each $b\in M$. In this paper, we prove that $f$ is a sequentially-quotient (resp. sequence-covering, compact-covering) mapping if and only if each $\mathcal {P}_n$ is a $cs^*$-cover (resp. $fcs$-cover, $cfp$-cover) of $X$. As a consequence of this result, we prove that $f$ is a sequentially-quotient, $s$-mapping if and only if it is a sequence-covering, $s$-mapping, where “$s$” can not be omitted.

Publié le : 2006-01-01
Classification:  54E40 
@article{108002,
     author = {Ying Ge},
     title = {On three equivalences concerning Ponomarev-systems},
     journal = {Archivum Mathematicum},
     volume = {042},
     year = {2006},
     pages = {239-246},
     zbl = {1164.54363},
     mrnumber = {2260382},
     language = {en},
     url = {http://dml.mathdoc.fr/item/108002}
}

Ge, Ying. On three equivalences concerning Ponomarev-systems. Archivum Mathematicum, Tome 042 (2006) pp. 239-246. http://gdmltest.u-ga.fr/item/108002/

Boone J. R.; And Siwiec F. Sequentially quotient mappings, Czechoslovak Math. J. 26 (1976), 174–182. (1976) | MR 0402689

Ge Y. On quotient compact images of locally separable metric spaces, Topology Proceedings 276 (2003), 351–560. | MR 2048944

Ge Y.; Gu J. On $\pi $-images of separable metric spaces, Mathematical Sciences 10 (2004), 65–71. | MR 2127483 | Zbl 1104.54014

Gruenhage G.; Michael E.; Tanaka Y. Spaces determined by point-countable covers, Pacific J. Math. 113 (1984), 303–332. (1984) | MR 0749538 | Zbl 0561.54016

Ikeda Y.; Liu C.; Tanaka Y. Quotient compact images of metric spaces and related matters, Topology Appl. 122 (2002), 237–252. | MR 1919303 | Zbl 0994.54015

Lin S. Point-countable covers and sequence-covering mappings, Chinese Science Press, Beijing, 2002. (Chinese) | MR 1939779 | Zbl 1004.54001

Lin S.; Yan P. Notes on $cfp$-covers, Comment. Math. Univ. Carolin. 44 (2003), 295–306. | MR 2026164 | Zbl 1100.54021

Michael E. $\aleph _0$-spaces, J. Math. Mech. 15 (1966), 983–1002. (1966) | MR 0206907

Ponomarev V. I. Axiom of countability and continuous mappings, Bull. Pol. Acad. Math. 8 (1960), 127–133. (1960) | MR 0116314

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