  Persistent homology is a popular tool in Topological Data Analysis. It
provides numerical characteristics of data sets which reflect global geometric
properties. In order to be useful in practice, for example for feature
generation in machine learning, it needs to be effectively computable.
Classical homology is a computable topological invariant because of the
Mayer-Vietoris exact and spectral sequences associated to coverings of a space.
We state and prove versions of the Mayer-Vietoris theorem for persistent
homology under mild and commonplace assumptions. This is done through the use
of a new theory, the singular persistent homology, better suited for handling
coverings of data sets. As an application, we create a distributed
computational workflow where the advantage is not only or even primarily in
speed improvement but in sheer feasibility for large data sets.
