Comments on "Vector Potentials in Simply Structured Spaces

In the recent paper of Kurokawa, a connected open space V is called simply structured if every closed surface contained in V forms the boundary of a bounded space contained in V, and a piecewise smooth divergence-free vector function of position is shown to have a vector potential in a simply structured space. On the other hand, if a connected open space is not simply structured, it is multiply structured and a piecewise smooth divergence-free vector function may or may not have a vector potential in a multiply structured space. So, one may wish to know when a vector potential exists in such a space. The following discussion answers this question.