In this paper it shown that all connected and connected im kleinen point sets lying in a separable metric space admit of decomposition into elements, called nodular elements, which are highly analogous to the cyclic elements of a continuous curvet and that the structure of these sets relative to their nodular elements very similar to the structure of continuous curves relative to their cyclic elements. Indeed, it true that for the special case of a continuous curve M, the nodular elements of are identically the same as the cyclic elements of M. The usual terminology and notation of point theory will be employed. To facilitate the reading of the paper, however, some of the less familiar terms and notations used are here defined and explained. The point P of a connected point a non-cut point or a cut point of according as -P or not connected. The subset X of a connected separates two subsets A and B of in provided that -X the sum of two mutually separated sets Ma(X) and Mb(X) containing A and B respectively, and this fact indicated by means of the equation MAX = Ma(X) +Mb(X). The subset K of a point said to be closed relative to M, or closed in M, provided that no point of -K a limit point of K. The symbol v means contains, and c means is contained in; 6(H) denotes the diameter of the point H, and p(X, Y) denotes the lower bound of the aggregate of numbers [p(x, y) ], where x and y are points of X and Y respectively and p(x, y) the distance from x to y. Since this paper concerned very largely with connected and connected im kleinen sets, it to be understood, unless definitely stated to the contrary, that when we speak of a set M it presupposed that a connected and connected im kleinen point set. The following results, some of which will be used later, either are already known or follow, as indicated below, quite readily from known properties of connected im kleinen point sets.