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Keywords:
Ponomarev-system; point-star network; $cs^*$-(resp. $fcs$-; $cfp$-)cover; sequentially-quotient (resp. sequence-covering; compact-covering) mapping
Ponomarev-system; point-star network; $cs^*$-(resp. $fcs$-; $cfp$-)cover; sequentially-quotient (resp. sequence-covering; compact-covering) mapping
Summary:
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.
References:
[1] Boone J. R., and Siwiec F.: Sequentially quotient mappings. Czechoslovak Math. J. 26 (1976), 174–182. MR 0402689
[2] Ge Y.: On quotient compact images of locally separable metric spaces. Topology Proceedings 276 (2003), 351–560. MR 2048944
[3] Ge Y., Gu J.: On $\pi $-images of separable metric spaces. Mathematical Sciences 10 (2004), 65–71. MR 2127483 | Zbl 1104.54014
[4] Gruenhage G., Michael E., Tanaka Y.: Spaces determined by point-countable covers. Pacific J. Math. 113 (1984), 303–332. MR 0749538 | Zbl 0561.54016
[5] 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
[6] Lin S.: Point-countable covers and sequence-covering mappings. Chinese Science Press, Beijing, 2002. (Chinese) MR 1939779 | Zbl 1004.54001
[7] Lin S., Yan P.: Notes on $cfp$-covers. Comment. Math. Univ. Carolin. 44 (2003), 295–306. MR 2026164 | Zbl 1100.54021
[8] Michael E.: $\aleph _0$-spaces. J. Math. Mech. 15 (1966), 983–1002. MR 0206907
[9] Ponomarev V. I.: Axiom of countability and continuous mappings. Bull. Pol. Acad. Math. 8 (1960), 127–133. MR 0116314
[10] 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
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