A theorem on open coverings of a simplex and Scarf's core existence theorem through Brouwer's fixed point theorem

Summary This paper provides simple proofs of a theorem on open coverings of a simplex and Scarf's core existence theorem through Brouwer's fixed point theorem.I want to thank Daniel Abrams, Yakar Kannai, Herbert Scarf, Nicholas Yannelis, Jingang Zhao, and an anonymous referee for their helpful comments. I also gratefully acknowledge the financial support from the Alfred P. Sloan Foundation.