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This paper deals with topical and sub-topical functions in a class of ordered Banach spaces. The separation theorem for downward sets and sub-topical functions is given. It is established some best approximation prob- lems by sub-topical functions and we will characterize sub-topical functions as superimum of elementary sub-topical functions.

SEPARATION THEOREM WITH RESPECT TO SUB-TOPICAL FUNCTIONS AND ABSTRACT CONVEXITY