Help me visualize this, boundary of the future

[JasonJo]

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?

 

[atyy]

I can visualise it only for various flat space scenarios. I googled it and these guys give an argument which seems to make sense, and they also say it's not obvious (Proposition 6.3, and Definition 25): http://books.google.com/books?id=zJ5rPOpiKjYC&printsec=frontcover#PPA85,M1

I think it's roughly like this. Suppose x is on the boundary of the future of S, and suppose y is in the future of x. Then x is in the past of y. The past of y is open, so there must be some open region surrounding x that is in the past of y. Because x is the boundary of the future of S, every open region surrounding x contains a point z in the future of S. So y is the future of an open region surrounding x containing some point z in the future of S. So y, the future of x, is in the future of S. So we don't have to worry that future of the boundary is also on the boundary.