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.
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.
METRIZATION
We start with a metrizable topological space. There are many equivalent axiomatizations of a metrizable topology; e.g., see Kelley. Perhaps the easiest way to visualize a metrizable topology is to consider that one was given a metric space but that he lost his notes in which the exact form of the metric was written down. Thus one knows that he can do everything that he could in a metric space, if only he can figure out how.
The “figuring out how” is by no means trivial. Here, it will be assumed that a cumulative probability distribution has been obtained on the space by one of the standard methods; bird in cage,[11] Munroe I,[12] Munroe II,[13] ordering (see Halmos[14] or Kelley[15]). This cumulative probability distribution is a function on X onto the interval [0,1] of real numbers. The inverse of this function, which exists by the Radon Nikodym theorem, provides a mapping from the real interval onto the non-trivial portion of X. This mapping induces all of the pleasant properties of the real numbers on the space X: topological, metric, and ordering.