Let S be any subset of M, where (M, g_ab) is a spacetime. Can you guys help me kind of visualize why the boundary of the chronological future of S is an achronal, 3 dimensional embedded manifold? I am just having a hard time seeing why this is so. I'm picturing a sphere, and then having all the null geodesics emanating from it. The null geodesics form the boundary of the chronological future. Why is this thing achronal? Why is it a submanifold?