In this chapter we shall establish some results that hold in ordered spaces in which light rays are complete (in which case they are locally homeomorphic with ℝ),^{1} but the space as a whole need not be order complete. We begin with an example of such a space, which is infinite-dimensional. We should add that we know of no example of a finite-dimensional ordered space in which light rays are complete but the space itself in not (order) complete. The phenomenon could be peculiar to infinite-dimensional spaces.