A "metrical" space is a manifold such that with any two points , is associated a real not-negative number having the following three properties: (a) ; (b) is only zero 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.