[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 "normal" when any two non-overlapping closed manifolds and can be separated by two non-overlapping regions , which respectively contain them and have no boundary-points. Ib., p. 310, and Hausdorff, op. cit., p. 215. A "boundary-point" of a collection is one which has a neighbourhood that is not a sub-class of the collection. [Pg 303] CHAPTER XXIX SPACE-TIME ORDER IN the present chapter I shall show how to develop spatio-temporal order, in the sense in which it is assumed by the general theory of relativity, without any apparatus beyond that of the preceding chapter, except a few hypotheses of the sort to be expected in founding analysis situs. The transformations of co-ordinates which are admissible in tensor analysis are not unlimited; they are such, only, as leave relations of neighbourhood unchanged.[66] That is to say, a small displacement in one system of co-ordinates must correspond to a small displacement in any other. This requires that, independently of metrical considerations, the events of the space-time manifold should have certain relations of order. It must be possible, in certain circumstances, to say that is nearer to than to , without presupposing any quantitative measure of distance. It must be possible to construct lines along which there is a definite order, but it must be impossible to distinguish certain lines as "straight." A closed curve will be distinguishable from an open curve, but two open curves will not be distinguishable from each other, provided they have no singularities. Generally, we shall be able to make propositions belonging to analysis situs, at any rate in a sufficiently small region. But propositions about a configuration must, in the geometry we are to construct, be only such as would remain true if the configuration were subjected to any kind of deformation which does not violate continuity. It is this pre-co-ordinate geometry that concerns us in the present chapter. [Pg 304] The order to be introduced is of two sorts, macroscopic and microscopic. We will treat first of the former.