The paper investigates the set of all selectively refined meshes that can be obtained from a progressive mesh. We call the set the transitive mesh space of a progressive mesh and present a theoretical analysis of the space. We define selective edge collapse and vertex split transformations, which we use to traverse all selectively refined meshes in the transitive mesh space. We propose a complete selective refinement scheme for a progressive mesh based on the transformations and compare the scheme with previous selective refinement schemes in both theoretical and experimental ways. In our comparison, we show that the complete scheme always generates selectively refined meshes with smaller numbers of vertices and faces than previous schemes for a given refinement criterion. The concept of dual pieces of the vertices in the vertex hierarchy plays a central role in the analysis of the transitive mesh space and the design of selective edge collapse and vertex split transformations.