Geodesic
Fragmentability of open sets via set-valued mappings
Serdica Mathematical Journal, Tome 44 (2018) no. 1-2, pp. 213-226

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We study the notion of fragmentability of nonempty open sets in a topological space X and provide several characterizations of this concept via properties of set-valued mappings taking their values in X. As a corollary we obtain that in a compact space X the nonempty open sets are fragmentable if, and only if, the set of all continuous real-valued functions in X which attain their minimum at exactly one point in X is a residual subset of the space C(X) of all bounded continuous real-valued functions in X, equipped with the uniform convergence norm.
Keywords: fragmentability of (open) sets, set-valued mappings, topological game, winning strategy, Baire category, strong minimum, 54E52, 54C60, 49J27, 49J53 
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     title = {Fragmentability of open sets via set-valued mappings},
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Choban, Mitrofan M.; Kenderov, Petar S.; Revalski, Julian P. Fragmentability of open sets via set-valued mappings. Serdica Mathematical Journal, Tome 44 (2018) no. 1-2, pp. 213-226. http://geodesic.mathdoc.fr/item/SMJ2_2018_44_1-2_a11/