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Lipschitz-type functions on metric spaces. (English) Zbl 1139.46025
Given a metric space $$(X,d)$$, let $$\text{Lip}(X)$$ and $$\text{Lip}^*(X)$$ denote, respectively, the set of all Lipschitz real functions and the set of all bounded Lipschitz real functions on $$(X,d)$$. In this paper, the authors examine the problem as to when $$\text{Lip}^*(X)$$ 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 $$f:(X,d) \to (Y,\rho)$$ between two metric spaces $$(X,d)$$ and $$(Y,\rho)$$ is Lipschitz in the small if there exist $$r>0$$ and $$K\geq 0$$ such that $$\rho(f(x),f(y)) \leq K\cdot d(x,y)$$ whenever $$d(x,y)\leq r$$. The space $$(X,d)$$ is called small-determined if $$\text{LS}(X)=\text{Lip}(X)$$. Further, $$(X,d)$$ and $$(Y,\rho)$$ 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 $$\text{LS}(X)$$ and $$\text{LS}(Y)$$ are isomorphic as unital vector lattices, if and only if $$\text{Lip}^*(X)$$ and $$\text{LS}(Y)$$ 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 $$\text{Lip}^*(X)$$ 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 in general topology 54C40 Algebraic properties of function spaces in general topology
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