We now define " is between and " as meaning: ", , are points such that is not null and is a proper part of ." An equivalent definition is: ", , are points such that is not null, and is contained in , but is not contained in ." By the help of suitable axioms, "between," so defined, can be made to give rise to the spatio-temporal order presupposed in assigning co-ordinates in the general theory of relativity. What the definition says, in geometrical language, is that every event which contains both and contains , but not every event which contains both and contains . We must not imagine that all the points between two others lie on one line; each lies on some short route joining the end-points, a "short" route being one composed wholly of points between the end-points; but none lies on all short routes. Before developing the formal consequences of this definition, it may be as well to consider its geometrical import. In the accompanying figure, will be between and if there are events which contain all three, but there are none which contain and without containing . (I represent events by areas.) Now if events can often be of irregular shapes[Pg 306] such as that of the shaded area in the figure, it would seem that one event is not likely ever to be between two others according to the definition. I shall therefore assume that we may picture events as free from re-entrant angles and similar oddities. I imagine them as all oval; but formally it would do just as well if they were all four-dimensional cubes, and it would not matter whether they were large or small, provided they did not differ too much, and were all above a certain minimum. These pictorial requisites are rather for the importance of the theory to be developed than for its truth. In the preceding chapter, we assumed that events are such as to be all spheres according to one possible metric. Formally, we might equally well have assumed that there is a metric in which they are all cubes. Some assumption of this kind, as we saw, is necessary for the success of our definition of points. The other assumptions needed for its truth will be explicitly stated as they are introduced. The assumptions introduced so far in this chapter and its predecessor are: (1) Compresence is symmetrical. (2) Defining "events" as the field of compresence, every event is compresent with itself. (3) Events can be well ordered; or at least those compresent with a given event can be. (4) Any two events have a relation which is a finite power of compresence. (This is required for mapping space-time into zones.) In other words, the ancestral relation derived from compresence is connected.