*(English)*Zbl 1139.46025

Given a metric space $(X,d)$, 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 $(X,d)$. 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 $f:(X,d)\to (Y,\rho )$ between two metric spaces $(X,d)$ and $(Y,\rho )$ is Lipschitz in the small if there exist $r>0$ and $K\ge 0$ such that $\rho \left(f\right(x),f(y\left)\right)\le K\xb7d(x,y)$ whenever $d(x,y)\le r$. The space $(X,d)$ is called small-determined if $\text{LS}\left(X\right)=\text{Lip}\left(X\right)$. 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 $\text{LS}\left(X\right)$ 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) |