The further development of our geometry, so as to include surfaces, volumes, and four-dimensional regions, obviously presents no difficulty in principle, and I do not propose to enlarge upon it. I will merely observe that it is possible to extend the method by which we have defined points and lines so as to obtain something which we may call surfaces and regions, though not quite in the usual sense. Probably various ways of doing this are possible; the one that I suggest is the following. A class of lines will be called "co-superficial" when any two intersect, but there is no point common to all the lines of the class. A "surface" is a co-superficial class of lines which cannot be augmented without ceasing to be co-superficial. A class of surfaces is "co-regional" when any two have a line in common, but no line is common to all the surfaces of the class. A "region" is a co-regional class of surfaces which cannot be augmented without ceasing to be co-regional. It is obvious that this method could be extended to any number of dimensions; also that it requires limitations and extensions. But it seems unnecessary to pursue the matter further, since it is plain that we have what is needed for the pre-co-ordinate geometry of space-time. Let us now compare our constructed space-time with the spatial manifolds of analysis situs. In the preceding chapter[Pg 312] we quoted Hausdorff's definition of a "topological" space, and we saw that, in order to prove the usual propositions about limits, it is necessary that the total number of neighbourhoods should be . Let us now define as a "neighbourhood" of a point any set of points each of which contains as a sub-class a certain finite co-punctual class of events which is a sub-class of . That is to say, if a is a co-punctual class of events each of which is a member of , the set of all the points of which a is a sub-class will be a neighbourhood of . With this definition of a "neighbourhood," it is obvious that our space has the four characteristics by which Hausdorff (loc. cit., p. 213) defines a topological space. In order to insure that our space shall also satisfy his second denumerative axiom (loc. cit., p. 263), it is necessary and sufficient to assume that the total number of events is . With this assumption, the theorems of analysis situs become applicable to our space-time manifold of points.