Given a metric space , let and denote, respectively, the set of all Lipschitz real functions and the set of all bounded Lipschitz real functions on . In this paper, the authors examine the problem as to when determines . 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 between two metric spaces and is Lipschitz in the small if there exist and such that whenever . The space is called small-determined if . Further, and are called LS-homeomorphic if there exists a homeomorphism such that and are Lipschitz in the small.
It is shown that two complete metric spaces and are LS-homeomorphic if and only if and are isomorphic as unital vector lattices, if and only if and are isomorphic as either algebras or unital vector lattices. Consequently, in the class of complete small-determined metric spaces , the Lip-structure of is determined by 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.