  Multidimensional persistence modules do not admit a concise representation
analogous to that provided by persistence diagrams for real-valued functions.
However, there is no obstruction for multidimensional persistent Betti numbers
to admit one. Therefore, it is reasonable to look for a generalization of
persistence diagrams concerning those properties that are related only to
persistent Betti numbers. In this paper, the persistence space of a
vector-valued continuous function is introduced to generalize the concept of
persistence diagram in this sense. The main result is its stability under
function perturbations: any change in vector-valued functions implies a not
greater change in the Hausdorff distance between their persistence spaces.
