  We reexamine the focusing effect crucial to the theorems that predict the
emergence of spacetime singularities and various results in the general theory
of black holes in general relativity. Our investigation incorporates the fully
nonlinear and dispersive nature of the underlying equations. We introduce and
thoroughly explore the concept of versal unfolding (topological normal form)
within the framework of the Newman-Penrose-Raychaudhuri system, the
convergence-vorticity equations (notably the first and third Sachs optical
equations), and the Oppenheimer-Snyder equation governing exactly spherical
collapse. The findings lead to a novel dynamical depiction of spacetime
singularities and black holes, exposing their continuous transformations into
new topological configurations guided by the bifurcation diagrams associated
with these problems.
