In this paper, we prove the two-dimensional case of a conjecture in general relativity, which states that if M is a nakedly singular future boundary or nakedly singular past boundary strongly causal spacetime, then the space of null geodesics, N\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathcal {N}$$\end{document}, 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. This fact implies the converse of the above conjecture; that is, if the space of null geodesics of a two-dimension causal continuous spacetime M is non-Hausdorff, then M is a nakedly singular future boundary or nakedly singular past boundary spacetime. However, some examples refute it for more dimensions.
