It is shown that by the use of Information Theory, any metrizable topology may be metrized as an orthogonal Euclidean space (with a random Gaussian probability distribution) times a denumerable random cartesian product of irreducible (wrt direct product) denumerable groups. The necessary algorithm to accomplish this metrization from a statistical basis is presented. If such a basis is unavailable, a certain nilpotent projection operator has to be used instead, as is shown in detail in the companion paper. This operator possesses self-organizing features. INTRODUCTION In the companion article[8] we will define a self-organizing system as one which, after observing the input and output of an unknown phenomenon (transfer relation), organizes itself into a simulation of the unknown phenomenon. [8] Kleyn, P. A., “Conceptual Design of Self-Organizing Machines,” Anaheim, California:Northrop Nortronics, NSS Report 2832, Nov. 14, 1963. Within the mathematical model, the aforementioned phenomenon may be represented as a topological space thus omitting for the moment the (arbitrary) designation of input and output which, as will be shown, bears on the question of uniqueness. Hence, for the purpose of this paper, which emphasizes the mathematical foundation, an intelligent device is taken as one which carries out the task of studying a space and describing it. In keeping with the policy that one should not ask someone (or something) else to do a task that he could not do himself (at least in principle), let us consider how we would approach such a problem. In the first place, we have to select the space in which the problem is to be set. The most general space that we feel capable of tackling is a metrizable topology. On the other hand, anything less general would be unnecessarily restrictive. Thus, we choose a metrizable topological space. As soon as we have made this choice, we regret it. In order to improve the situation somewhat, we show that there is no (additional) loss of generality in using an orthogonal Euclidean space times[9] a denumerable random cartesian product of irreducible (wrt direct product) denumerable groups. This paper provides a survey of the problem and a method for solving it which is conceptually clear but not very practical. The companion paper[10] provides a practical method for solving this problem by means of the successive use of a certain nilpotent projection operator. [9] Random cartesian product. [10] Kleyn, P. A., “Conceptual Design of Self-Organizing Machines,” Anaheim, California:Northrop Nortronics, NSS Report 2832, Nov. 14, 1963. METRIZATION