The Set of Distances in a Polish Metric Space

In this article we consider the possible sets of distances in Polish metric spaces. By a Polish metric space we mean a pair (X, d), where X is a Polish space (a separable, completely-metrizable space) and d is a complete, compatible metric for X. We will consider two aspects. First, we will characterize which sets of reals can be the set of distances in a Polish metric space. We will also obtain results about specific classes of metric spaces, e.g. compact, locally compact, zero-dimensional. In the last section we will briefly consider the question of which sets of triangles (and larger configurations) can occur in a Polish metric space. Our interest in distance sets is somewhat motivated by the question of classifying metric spaces up to isometry, since distance sets form an isometry invariant, although generally not a complete invariant.