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 Mat. Sb., 2006, Volume 197, Number 9, Pages 3–18 (Mi msb1129)

Monotone path-connectedness of Chebyshev sets in the space $C(Q)$

A. R. Alimov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The structure of Chebyshev sets and strict suns in the space $C(Q)$ with compact $Q$ is considered. It is shown that a boundedly compact strict sun in $C(Q)$ (in particular, a bounded compact Chebyshev set) is monotone path-connected and in particular, $P$-acyclic. It is demonstrated that a monotone path-connected Chebyshev set in $C(Q)$ is a sun.
Bibliography: 25 titles.

DOI: https://doi.org/10.4213/sm1129

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English version:
Sbornik: Mathematics, 2006, 197:9, 1259–1272

Bibliographic databases:

UDC: 517.982.256
MSC: Primary 41A65; Secondary 46B20

Citation: A. R. Alimov, “Monotone path-connectedness of Chebyshev sets in the space $C(Q)$”, Mat. Sb., 197:9 (2006), 3–18; Sb. Math., 197:9 (2006), 1259–1272

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb1129
• https://doi.org/10.4213/sm1129
• http://mi.mathnet.ru/eng/msb/v197/i9/p3

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. R. Alimov, “A Monotone Path Connected Chebyshev Set Is a Sun”, Math. Notes, 91:2 (2012), 290–292
2. A. R. Alimov, “Monotone path-connectedness of $R$-weakly convex sets in the space $C(Q)$”, J. Math. Sci., 185:3 (2012), 360–366
3. A. R. Alimov, V. Yu. Protasov, “Separation of convex sets by extreme hyperplanes”, J. Math. Sci., 191:5 (2013), 599–604
4. A. R. Alimov, “Monotone path-connectedness of $R$-weakly convex sets in spaces with linear ball embedding”, Eurasian Math. J., 3:2 (2012), 21–30
5. A. R. Alimov, “Bounded strict solar property of strict suns in the space $C(Q)$”, Moscow University Mathematics Bulletin, 68:1 (2013), 14–17
6. A. R. Alimov, “Local solarity of suns in normed linear spaces”, J. Math. Sci., 197:4 (2014), 447–454
7. A. R. Alimov, “Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces”, Izv. Math., 78:4 (2014), 641–655
8. A. R. Alimov, “The Rainwater–Simons weak convergence theorem for the Brown associated norm”, Eurasian Math. J., 5:2 (2014), 126–131
9. 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
10. A. R. Alimov, “On finite-dimensional Banach spaces in which suns are connected”, Eurasian Math. J., 6:4 (2015), 7–18
11. 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
12. I. G. Tsar'kov, “Continuous $\varepsilon$-Selection and Monotone Path-Connected Sets”, Math. Notes, 101:6 (2017), 1040–1049
13. A. R. Alimov, “Selections of the metric projection operator and strict solarity of sets with continuous metric projection”, Sb. Math., 208:7 (2017), 915–928
14. Tsar'kov I.G., “Continuous Selection From the Sets of Best and Near-Best Approximation”, Dokl. Math., 96:1 (2017), 362–364
15. A. R. Alimov, “A monotone path-connected set with outer radially lower continuous metric projection is a strict sun”, Siberian Math. J., 58:1 (2017), 11–15
16. Alimov A.R., “On Approximative Properties of Locally Chebyshev Sets”, Proc. Inst. Math. Mech., 44:1 (2018), 36–42
17. I. G. Tsar'kov, “Continuous selections for metric projection operators and for their generalizations”, Izv. Math., 82:4 (2018), 837–859
18. I. G. Tsar'kov, “New Criteria for the Existence of a Continuous $\varepsilon$-Selection”, Math. Notes, 104:5 (2018), 727–734
19. A. R. Alimov, “Selections of the best and near-best approximation operators and solarity”, Proc. Steklov Inst. Math., 303 (2018), 10–17
20. I. G. Tsar'kov, “Weakly monotone sets and continuous selection from a near-best approximation operator”, Proc. Steklov Inst. Math., 303 (2018), 227–238
21. Alimov A.R., “Continuity of the Metric Projection and Local Solar Properties of Sets: Continuity of the Metric Projection and Solar Properties”, Set-Valued Var. Anal., 27:1 (2019), 213–222
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