 So far, all we have defined is a set of events. To be a four-dimensional manifold, the set of events must be a little bit more organized. In a real spacetime, we have the idea that each event sits in some local neighborhood of events; and this neighborhood sits inside a larger neighborhood of events; and so on. That extra organization comes from the requirement that we can smoothly label the events with four numbers—or at least we can do this for any sufficiently small chunk of the manifold. These labels form coordinate systems. The fact that four numbers are just sufficient to label the events makes the manifold four-dimensional. We can now pick out neighborhoods of some event as the set of all points whose spacetime coordinates differ from our starting event by at most one unit; or two units; or three units; etc.. That gives us the nestling neighborhoods of events.  Figure 1 illustrates how a set of events may be made into a two dimensional manifold by assigning “x” and “y” coordinates to the events.