Generating continuous mappings with Lipschitz mappings

Abstract

If X is a metric space, then C X and £ X denote the semigroups of continuous and Lipschitz mappings, respectively, from X to itself. The relative rank of C X modulo £ X is the least cardinality of any set U £ X where U generates C X . For a large class of separable metric spaces X we prove that the relative rank of C X modulo £ X is uncountable. When X is the Baire space N N , this rank is 8 1 . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.