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Lipschitz-type functions on metric spaces. (English) Zbl 1139.46025

Given a metric space $\left(X,d\right)$, let $\text{Lip}\left(X\right)$ and ${\text{Lip}}^{*}\left(X\right)$ denote, respectively, the set of all Lipschitz real functions and the set of all bounded Lipschitz real functions on $\left(X,d\right)$. In this paper, the authors examine the problem as to when ${\text{Lip}}^{*}\left(X\right)$ determines $X$. To achieve their goal, they introduce several types of Lipschitz functions. In particular, Lipschitz functions in the small and small-determined spaces are found to be useful. Recall that a function $f:\left(X,d\right)\to \left(Y,\rho \right)$ between two metric spaces $\left(X,d\right)$ and $\left(Y,\rho \right)$ is Lipschitz in the small if there exist $r>0$ and $K\ge 0$ such that $\rho \left(f\left(x\right),f\left(y\right)\right)\le K·d\left(x,y\right)$ whenever $d\left(x,y\right)\le r$. The space $\left(X,d\right)$ is called small-determined if $\text{LS}\left(X\right)=\text{Lip}\left(X\right)$. Further, $\left(X,d\right)$ and $\left(Y,\rho \right)$ are called LS-homeomorphic if there exists a homeomorphism $h$ such that $h$ and ${h}^{-1}$ are Lipschitz in the small.

It is shown that two complete metric spaces $X$ and $Y$ are LS-homeomorphic if and only if $\text{LS}\left(X\right)$ and $\text{LS}\left(Y\right)$ are isomorphic as unital vector lattices, if and only if ${\text{Lip}}^{*}\left(X\right)$ and $\text{LS}\left(Y\right)$ are isomorphic as either algebras or unital vector lattices. Consequently, in the class of complete small-determined metric spaces $X$, the Lip-structure of $X$ is determined by ${\text{Lip}}^{*}\left(X\right)$ as an algebra or a unital vector lattice. This is a theorem of the Banach-Stone type. Small-determined metric spaces are LS-homeomorphic invariants, and the class of small-determined spaces includes bounded weakly precompact metric spaces, as well as quasi-convex metric spaces. The authors also investigate properties of small-determined spaces. Among many other things, it is shown that small-determined spaces are precisely those metric spaces on which every uniformly continuous real function can be uniformly approximated by Lipschitz functions.

##### MSC:
 46E05 Lattices of continuous, differentiable or analytic functions 54E35 Metric spaces, metrizability 54C35 Function spaces (general topology) 54C40 Algebraic properties of function spaces (general topology)