  We study the geometry of curves in the Minkowski space and in the de Sitter
space, specially at points where the tangent direction is lightlike (i.e. has
length zero) called lightlike points of the curve. We define the focal sets of
these curves and study the metric structure of them. At the lightlike points,
the focal set is not defined. We use singularity theory techniques to carry out
our study and investigate the focal set near lightlike points.
