  In this paper, We prove a conjecture which states that if M is a nakedly
singular future boundary or nakedly singular past boundary spacetime, then the
space of null geodesics, N, is non-Hausdorff. Also, we show that every
two-dimensional strongly causal spacetime M is causally simple if and only if
it is null pseudoconvex. As a result, it implies the converse of the conjecture
for two-dimension but there are examples that refute it for more dimensions.
