Generating continuous mappings with Lipschitz mappings

Abstract:

If $X$ is a metric space, then $\mathcal {C}_{X}$ and $\mathcal {L}_X$ denote the semigroups of continuous and Lipschitz mappings, respectively, from $X$ to itself. The relative rank of $\mathcal {C}_{X}$ modulo $\mathcal {L}_{X}$ is the least cardinality of any set $U\setminus \mathcal {L}_{X}$ where $U$ generates $\mathcal {C}_{X}$. For a large class of separable metric spaces $X$ we prove that the relative rank of $\mathcal {C}_{X}$ modulo $\mathcal {L}_X$ is uncountable. When $X$ is the Baire space $\mathbb {N}^{\mathbb {N}}$, this rank is $\aleph _{1}$. 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.