A homeomorphism f: (X, d) →(X, d) of a metric space (X, d) onto X is recurrent provided that for each ε > 0 there exists a positive integer n such that fn is ε-close to the identity map on X. The notion of a recurrent homeomorphism is weaker than that of an almost periodic homeomorphism. The result announced in the title generalizes the theorem of Brechner for almost periodic homeomorphisms and answers a question of R. D. Edwards. © 1990 American Mathematical Society.