We introduce the concept of a re nable set relative to a family of contractive mappings on a metric space, and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure. The notion of a re nable set parallels that of a re nable function, which is the basis of wavelet construction. The interpolation points we recursively generate from a re nable set by a set-theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction. We then use this recursive structure for the points to construct multiscale interpolants. Several concrete examples of re nable sets which can be used for generating interpolatory wavelets are included.