Abstract
We address the approximative and structural properties of approximating sets in asymmetric spaces. More precisely, we study the interrelations between the new concept of \( \overset{\circ}{B} \)-connected set and a few classical structural characteristics of sets, in particular, we examine whether \( \overset{\circ}{B} \)-complete sets have connected or path-connected intersections with closed and open balls. A \( \overset{\circ}{B} \)-complete Chebyshev set in an asymmetric Efimov–Stechkin space is shown to be \( B \)-connected, i.e., it has connected intersections with closed balls. All problems under consideration are posed in asymmetric and classical normed spaces.
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Notes
Here \( \overset{\circ}{B} \) stands for the open unit ball.
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Acknowledgment
The authors are grateful to the referee for many useful remarks.
Funding
This research is supported by the Russian Science Foundation under Grant no. 22–21–00204.
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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 3, pp. 500–509. https://doi.org/10.33048/smzh.2022.63.302
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Alimov, A.R., Tsarkov, I.G. \( \overset{\circ}{B} \)-Complete Sets: Approximative and Structural Properties. Sib Math J 63, 412–420 (2022). https://doi.org/10.1134/S0037446622030028
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DOI: https://doi.org/10.1134/S0037446622030028
Keywords
- approximation in asymmetric spaces
- \( \overset{\circ}{B} \)-complete set
- \( B \)-connected set
- asymmetric Efimov–Stechkin space
- Chebyshev set