On three equivalences concerning Ponomarev-systems

Ying Ge

Archivum Mathematicum (2006)

  • Volume: 042, Issue: 3, page 239-246
  • ISSN: 0044-8753

Abstract

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Let { 𝒫 n } be a sequence of covers of a space X such that { s t ( x , 𝒫 n ) } is a network at x in X for each x ∈ X . For each n ∈ ℕ , let 𝒫 n = { P β : β ∈ Λ n } and Λ n be endowed the discrete topology. Put M = { b = ( β n ) ∈ Π n ∈ ℕ Λ n : { P β n } forms a network at some point x b i n X } and f : M ⟶ X by choosing f ( b ) = x b for each b ∈ M . In this paper, we prove that f is a sequentially-quotient (resp. sequence-covering, compact-covering) mapping if and only if each 𝒫 n is a c s * -cover (resp. f c s -cover, c f p -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.

How to cite

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Ge, Ying. "On three equivalences concerning Ponomarev-systems." Archivum Mathematicum 042.3 (2006): 239-246. <http://eudml.org/doc/249820>.

@article{Ge2006,
abstract = {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.},
author = {Ge, Ying},
journal = {Archivum Mathematicum},
keywords = {Ponomarev-system; point-star network; $cs^*$-(resp. $fcs$-; $cfp$-)cover; sequentially-quotient (resp. sequence-covering; compact-covering) mapping; Ponomarev-system; point-star network},
language = {eng},
number = {3},
pages = {239-246},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On three equivalences concerning Ponomarev-systems},
url = {http://eudml.org/doc/249820},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Ge, Ying
TI - On three equivalences concerning Ponomarev-systems
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 3
SP - 239
EP - 246
AB - 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.
LA - eng
KW - Ponomarev-system; point-star network; $cs^*$-(resp. $fcs$-; $cfp$-)cover; sequentially-quotient (resp. sequence-covering; compact-covering) mapping; Ponomarev-system; point-star network
UR - http://eudml.org/doc/249820
ER -

References

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  5. Ikeda Y., Liu C., Tanaka Y., Quotient compact images of metric spaces and related matters, Topology Appl. 122 (2002), 237–252. Zbl0994.54015MR1919303
  6. Lin S., Point-countable covers and sequence-covering mappings, Chinese Science Press, Beijing, 2002. (Chinese) Zbl1004.54001MR1939779
  7. Lin S., Yan P., Notes on c f p -covers, Comment. Math. Univ. Carolin. 44 (2003), 295–306. Zbl1100.54021MR2026164
  8. Michael E., ℵ 0 -spaces, J. Math. Mech. 15 (1966), 983–1002. (1966) MR0206907
  9. Ponomarev V. I., Axiom of countability and continuous mappings, Bull. Pol. Acad. Math. 8 (1960), 127–133. (1960) MR0116314
  10. Tanaka Y., Ge Y., Around quotient compact images of metric spaces and symmetric spaces, Houston J. Math. 32 (2006), 99–117. Zbl1102.54034MR2202355

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