Geodesic
Weak selections and weak orderability of function spaces
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 273-281

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It is proved that for a zero-dimensional space $X$, the function space $C_p(X,2)$ has a Vietoris continuous selection for its hyperspace of at most 2-point sets if and only if $X$ is separable. This provides the complete affirmative solution to a question posed by Tamariz-Mascarúa. It is also obtained that for a strongly zero-dimensional metrizable space $E$, the function space $C_p(X,E)$ is weakly orderable if and only if its hyperspace of at most 2-point sets has a Vietoris continuous selection. This provides a partial positive answer to a question posed by van Mill and Wattel.
Classification : 54B20, 54C35, 54C65, 54F05
Keywords: Vietoris hyperspace; continuous selection; function space; weakly orderable space 
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Gutev, Valentin. Weak selections and weak orderability of function spaces. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 273-281. http://geodesic.mathdoc.fr/item/CMJ_2010__60_1_a21/