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. We study the relations of such functions with topometric versions of classical separation axioms, namely, normality and complete regularity, as well as with completions of topometric spaces. We also recover a compact topometric space $X$ from the lattice of continuous $1$-Lipschitz functions on $X$, in analogy with the recovery of a compact topological space $X$ from the structure of (real or complex) functions on $X$.

Mots clés: topometric space; Lipschitz function; normal; completely regular; Urysohn's Lemma; Stone-Čech compactification; completion; function space; lattice

Domaines: Logique [math.LO]

Itaï Ben Yaacov. Lipschitz functions on topometric spaces. 2013. ⟨hal-00524416v2⟩