  Phase singularities as topological objects of wave fields appear in a variety
of physical, chemical, and biological scenarios. In this paper, by making use
of the $\phi$-mapping topological current theory, we study the topological
properties of the phase singularities in two and three dimensional space in
details. The topological inner structure of the phase singularities are
obtained, and the topological charge of the phase singularities are expressed
by the topological numbers: Hopf indices and Brouwer degrees. Furthermore, the
topological invariant of the closed and knotted phase singularities in three
dimensional space are also discussed in details.
