Lipschitz functions on topometric spaces

Details Lipschitz functions on topometric spaces Lipschitz functions on topometric spaces

2010-10-08

arxiv.org/abs/1010.1600v1

Mathematics

Logic

Scientific paper

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. (1) We define \emph{normal} topometric spaces and characterise them by analogues of Urysohn's Lemma and Tietze's Extension Theorem. (2) We define \emph{completely regular} topometric spaces and characterise them by the existence of a topometric Stone-\v{C}ech compactification. (3) 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 $X$.

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