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Izv. RAN. Ser. Mat., 2005, Volume 69, Issue 4, Pages 3–18 (Mi izv645)  

This article is cited in 12 scientific papers (total in 12 papers)

Connectedness of suns in the space $c_0$

A. R. Alimov


Abstract: We study the question of the connectedness of suns in the space $c_0$. We show that any sun $M$ in $c_0$ is m-connected (in the sense of Brown). It follows that $M$ is monotonically path-connected and the intersection of $M$ with an arbitrary ball in $c_0$ is monotonically path-connected (and, in particular, path-connected). On the other hand, we establish that every approximatively compact m-connected set in $c_0$ is a sun in $c_0$. For $X=c_0$, $c$ or $\ell^\infty$, it is proved that the intersection of a sun in $X$ with a finite-dimensional coordinate subspace $H_n\subset X$ is a $P$-acyclic sun in $H_n$.

DOI: https://doi.org/10.4213/im645

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English version:
Izvestiya: Mathematics, 2005, 69:4, 651–666

Bibliographic databases:

UDC: 517.982.256
MSC: 41A65
Received: 31.05.2004

Citation: A. R. Alimov, “Connectedness of suns in the space $c_0$”, Izv. RAN. Ser. Mat., 69:4 (2005), 3–18; Izv. Math., 69:4 (2005), 651–666

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. R. Alimov, “Geometric construction of Chebyshev sets in the spaces $\ell^\infty(n)$, $c_0$ and $c$”, Russian Math. Surveys, 60:3 (2005), 559–561  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. R. Alimov, “Preservation of approximative properties of subsets of Chebyshev sets and suns in $\ell^\infty (n)$”, Izv. Math., 70:5 (2006), 857–866  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. A. R. Alimov, “Monotone path-connectedness of Chebyshev sets in the space $C(Q)$”, Sb. Math., 197:9 (2006), 1259–1272  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. A. R. Alimov, “A Monotone Path Connected Chebyshev Set Is a Sun”, Math. Notes, 91:2 (2012), 290–292  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. A. R. Alimov, “Monotone path-connectedness of $R$-weakly convex sets in the space $C(Q)$”, J. Math. Sci., 185:3 (2012), 360–366  mathnet  crossref
    6. A. R. Alimov, “Local solarity of suns in normed linear spaces”, J. Math. Sci., 197:4 (2014), 447–454  mathnet  crossref
    7. A. R. Alimov, “Bounded strict solar property of strict suns in the space $C(Q)$”, Moscow University Mathematics Bulletin, 68:1 (2013), 14–17  mathnet  crossref
    8. A. R. Alimov, “Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces”, Izv. Math., 78:4 (2014), 641–655  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. A. R. Alimov, “The Rainwater–Simons weak convergence theorem for the Brown associated norm”, Eurasian Math. J., 5:2 (2014), 126–131  mathnet
    10. A. R. Alimov, I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci., 217:6 (2016), 683–730  mathnet  crossref  mathscinet
    11. A. R. Alimov, “On finite-dimensional Banach spaces in which suns are connected”, Eurasian Math. J., 6:4 (2015), 7–18  mathnet
    12. A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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