A B S T R A C TBy generalizing the classical Knaster-Kuratowski-Mazurkiewicz Theorem, we obtain a result that provides su¢cient conditions to ensure the non-emptiness of several kinds of choice functions. This result generalizes well-known results on the existence of maximal elements for binary relations (Bergstrom, 1975;Walker, 1977;Tian, 1993), on the non-emptiness of nonbinary choice functions (Nehring, 1996;Llinares and Sánchez, 1999) and on the non-emptiness of some classical solutions for tournaments (top cycle and uncovered set) on non-…nite sets.

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