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In this paper we deal with weakly upper semi-continuous set-valued maps, taking arbitrary non-empty values, from a non-metric domain to a Banach space. We obtain selectors having the point of continuity property relative to the norm topology for a large class of compact spaces as a domain. Exact conditions under which the selector is of the first Borel class are also investigated.

Generalized first class selectors for upper semi-continuous set-valued maps in Banach spaces