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This paper was extensively circulated in manuscript form beginning
in the Summer of 1989. It is being published here for the first time in
its original form except for minor corrections, updated references and some
concluding comments.
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.
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