The Analysis of Matter

when and are identical; (c) is greater than or equal to .[62] A "topological" space is a manifold whose elements are associated with sub-classes of the manifold such that: (A) To every corresponds at least one , and every contains ; (B) If , are both neighbourhoods of there is a neighbourhood of , say which is contained in the common part of and ; (C) If y is a member of , there is a neighbourhood of which is contained in ; (D) Given any two distinct points, there is a neighbourhood of the one and there is a neighbourhood of the other such that the two have no common point.[63] [Pg 297] In order to be able to apply the usual methods of limits to a topological space, Hausdorff has need of an "Abzählbarkeitsaxiom," or "denumerative axiom." He gives two such axioms (p. 263), of which the first is the weaker, and is for some purposes insufficient. The first states that the number of neighbourhoods of a given point is never greater than ; the second states that the total number of neighbourhoods of all points is together . This second axiom suffices for all the usual kinds of argument, without the introduction of any metrical ideas. P. Urysohn[64] has shown that every topological space which satisfies Hausdorff's second denumerative axiom and has one further property (which he calls "normality"[65]) is metricizable. These are the main points from analysis situs that are relevant to the solution of our problem. For the present, we are not concerned with metrical properties, but only with such as belong to "topological" spaces. In virtue of Urysohn's theorem, it will be possible to introduce a metric if we can construct the right sort of topological space. But when one metric is possible, an infinite number are possible. The metric which is actually introduced in theory of relativity is introduced for empirical reasons; it uses a quantitative relation which might be called degree of causal proximity. The existence of this relation is not implied by anything with which we are at present concerned. Moreover, the metrical manifold which we require in physics is not a "metrical space" according to Hausdorff's definition given above, since interval in relativity does not possess the properties (b) and (c)[Pg 298] which distance possesses in Hausdorff's definition. However, so far as topological considerations are concerned, we may, without appreciable inaccuracy, assign to small regions the topological properties which belong to a small region of Euclidean space lasting for a short time, i.e. to a continuous series of small regions of Euclidean space all geometrically indistinguishable. In analysis situs, both points and neighbourhoods are given. We, on the other hand, wish to define our points in terms of "events" where "events" will have a one-one correspondence with certain neighbourhoods. We want our "events" to correspond with neighbourhoods which are above a certain minimum and below a certain maximum when, at a later stage, the empirical metric is introduced. We have to assign to our events such properties as will enable us to define the points of a topological space as classes of events, and the neighbourhoods of the points as classes of points. But we have to remember that we do not want to construct merely a topological space: what we want to construct is the four-dimensional space-time of the general theory of relativity. The following illustration will serve to introduce the problem. Consider a three-dimensional Euclidean numerical space, i.e. the manifold of all ordered triads of real numbers (, , ), with the usual definition of distance. Consider, in this space, all the spheres having a given radius and having centres whose co-ordinates are rational. The number of such spheres is . Let us define a group of these spheres as "co-punctual" if it is such that every four chosen out of the group have a common region; and let us define a co-punctual group as "punctual" if it cannot be enlarged without ceasing to be co-punctual. Then there is a one-one correspondence between the original points of our space and the punctual groups of spheres. Consequently the punctual groups of spheres form a Euclidean space. If the spheres are all distorted in any continuous way,[Pg 299] they will still enable us to construct punctual groups in the same way, and the manifold of punctual groups will still have all the topological properties which are possessed by a three-dimensional Euclidean space. Therefore if we are to use this method of constructing points out of "events," we shall have to assume that, in the resulting space, there is a possible metric according to which the points of which a given event is a member always form a spherical volume. Although this is expressed in metrical language, it is in reality a topological property, since it is unaffected by continuous deformation. It must be possible to express it in non-metrical language, though I must confess that I lack the necessary skill. I propose, therefore, to regard events as occupying regions of space-time which, in some possible metric, are spheres so far as their space-dimensions are concerned, and between a certain maximum and a certain minimum so far as their time-dimension is concerned. The region "occupied" by an event is the class of points of which it is a member. As the fundamental relation in the construction of points, we take a five-term relation of "co-punctuality," which holds between five events when there is a region common to all of them. A group of five or more events is called "co-punctual" when every quintet chosen out of the group has the relation of co-punctuality. A "point" is a co-punctual group which cannot be enlarged without ceasing to be co-punctual. In order to demonstrate the existence of points so defined, it is sufficient to assume that all events (or at least all events co-punctual with a given co-punctual quintet) can be well ordered. If Zermelo's axiom is true, this must be the case; if not, it may involve some limitation as to the number of events. I have been led by the arguments, first of Dr H. M. Sheffer, and then of Mr F. P. Ramsey, to the view that Zermelo's axiom is true; I am therefore less reluctant than I[Pg 300] should have been formerly to assume that events can be well ordered. To prove that every event is a member of at least one point, we proceed as follows—assuming that there are co-punctual quintets. Let be a well-ordered series whose field consists of all events; put Let , , , , be a co-punctual quintet. If is the only event co-punctual , , , , then the class whose only members are , , , , is a point according to the definition. If, on the other hand, there are 's other than which are co-punctual with , , , , , let be the first of them. If no other than and is co-punctual with , , , , and , then , , , , and form a point. Otherwise, let be the first other than and and co-punctual with , , , , , , then must be later in the -series than . If this process comes to an end with , then , , , , , , ... together form a point. If it does not come to an end with any finite , it may happen that no outside the series (, , ... ,...) is co-punctual with , , , and all the 's; in that case, , , , and these 's form a point. But if there are 's other than the 's and co-punctual with all of them, let be the first of them. Then is later in the -series than any of the finite 's. We proceed in this way as long as possible, using two principles: (1) given a series of 's ending with , let be the first in the -series after and co-punctual with the group of all the previous 's; (2) given a series of 's having no last term, take as the next the first in the -series which is after all the 's hitherto selected and co-punctual with all of them. If, at any stage, there is no such , the 's already selected form a point. Now this process must end sooner or later; for the 's (other than ) form an ascending series[Pg 301] selected from , and therefore, sooner or later, there will be no 's later than all the 's previously selected. At this stage, if not before, , , , and the 's already selected will form a point. Hence if all events can be well ordered, every event is a member of at least one point, provided every event is a member of a co-punctual quintet. The proof still holds if we only assume that all events co-punctual with a given quintet can be well ordered. Given any class of events , let be the class of those events which are co-punctual with a. Then by definition a is a point if . The necessary and sufficient condition that all the members of a should have a point in common is that a should be contained in . This condition is necessary, for, if is a point and is contained in , it follows that is contained in , and that , so that is contained in . The proof that the condition is sufficient is longer; it is as follows. If , is a point. If not, let denote the part of which is outside . Using again the -series of all events, put and so on, as long as possible. If , precedes in the -order. Hence, as before, there must come a stage when no fresh 's can be constructed. If is the class consisting of a together with all the 's yielded by the method, is a point. For (1) all the quintets in are co-punctual, by the construction; (2) a term co-punctual with all the quartets of cannot be later than all the 's, because if there were such a term we could construct more 's; (3) such a term cannot be[Pg 302] earlier than some member of because, if it were, it would have been chosen as the of that stage in the construction; hence no event outside is co-punctual with every quartet of . Hence is a point. To say that a collection of events have a point in common is to say that the collection is part (or the whole) of the class which is the point. Conversely, a collection of events may contain a sub-class which is a point; the necessary and sufficient condition for this is that should be contained in , where is the collection in question. The proof proceeds exactly as before, if we now make mean the part of which is not contained in . A group of events a is "co-punctual" if is contained in , and a "point" is a co-punctual group which cannot be enlarged without ceasing to be co-punctual. A few purely logical properties of points may be noted. Given any two classes and , if is contained in , then is contained in . Hence if and are points and is contained in , and are identical; for in that case and are respectively identical with and , and therefore if is contained in , is contained in , so that and are identical. Every co-punctual group of events contains at least one point. This has already been proved, since to say that a is a co-punctual group is to say that a is contained in . It may be taken that, in general, there are a number of points of which any given event is a member. Such a set of points will fill a "region," but not every region will be the set of points to which some one event belongs. This topic, however, cannot be dealt with until we have discussed space-time order. FOOTNOTES: [59] In this chapter and the next, I owe much to the criticism and suggestions of Mr M. H. A. Newman of St. John's College, Cambridge, who must not, however, be held responsible for their contents; on me contrary, I am convinced that he could construct a much better theory than that which follows. [60] Stetige Mengen, Monatshafte für Mathematik u. Physik., XXXI., 1921, pp. 173-204. [61] Grundzüge der Mengenlehre, Leipzig, 1914. [62] Ib., p. 211. [63] Ib., p. 213. [64] Zum Metrisationsproblem, Math. Annalen 94 (1925), pp. 309-315. [65] He defines a topological space as "