Lipschitz functions on topometric spaces

Itaï Ben Yaacov (1)

1 ICJ - Institut Camille Jordan

We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used. \begin{enumerate} \item We define \emph{normal} topometric spaces and characterise them by analogues of Urysohn's Lemma and Tietze's Extension Theorem. \item We define \emph{completely regular} topometric spaces and characterise them by the existence of a topometric Stone-Čech compactification. \item For a compact topological space $X$, we characterise the subsets of $\cC(X)$ which can arise as the set of continuous $1$-Lipschitz functions with respect to a topometric structure on $X$.

Mots clés: topometric space; normal topometric space; completely regular topometric space}

Domaines: Logique [math.LO]

Itaï Ben Yaacov. Lipschitz functions on topometric spaces. 2010. ⟨hal-00524416v1⟩