Geodesic
Weak selections and flows in networks
Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 3, pp. 509-517.

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We demonstrate that every Vietoris continuous selection for the hyperspace of at most 3-point subsets implies the existence of a continuous selection for the hyperspace of at most 4-point subsets. However, in general, we do not know if such ``extensions'' are possible for hyperspaces of sets of other cardinalities. In particular, we do not know if the hyperspace of at most 3-point subsets has a continuous selection provided the hyperspace of at most 2-point subsets has a continuous selection.
Classification : 05C90, 05C99, 54B20, 54C65
Keywords: hyperspace topology; Vietoris topology; continuous selection; flow; network 
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Gutev, Valentin; Nogura, Tsugunori. Weak selections and flows in networks. Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 3, pp. 509-517. http://geodesic.mathdoc.fr/item/CMUC_2008__49_3_a10/