Geodesic
Local and global continuous $\varepsilon$-selection
Izvestiya. Mathematics , Tome 80 (2016) no. 2, pp. 442-461.

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We study properties of sets for which there is a continuous selection from the set of almost-best approximants. We establish relations between local and global selections, give various examples of sets possessing a continuous $\varepsilon$-selection, and introduce the notions of moduli of approximative continuity, approximative $\delta$-solarity and uniform approximative continuity. These notions enable us to establish the $\delta$-solarity of sets under certain conditions.
Keywords: $\varepsilon$-selection, monotone path-connected sets, $\delta$-solarity, $\mathring{B}$-infinite connectedness, $\mathring{B}$-approximative infinite connectedness, moduli of approximative continuity. 
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I. G. Tsar'kov. Local and global continuous $\varepsilon$-selection. Izvestiya. Mathematics , Tome 80 (2016) no. 2, pp. 442-461. http://geodesic.mathdoc.fr/item/IM2_2016_80_2_a8/

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