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. Actually, it turns out that, especially if the dimensionality of the space is greater than one, the foregoing procedure not only provides one metrization, but many. Indeed, this lack of uniqueness is what makes the procedure exceedingly difficult. Only by imposing some additional conditions that result in the existence of a unique solution, does the problem become tractable. We choose to impose the additional condition that the resulting metric space be a Euclidean geometry with a rectangular coordinate system. [11] Harman, W. W., “Principles of the Statistical Theory of Communication,” New York, New York:McGraw-Hill, 1963. [12] Munroe, M. E., “Introduction to Measure and Integration,” Cambridge, Mass.:Addison-Wesley, 1953. [13] Munroe, M. E., “Introduction to Measure and Integration,” Cambridge, Mass.:Addison-Wesley, 1953. [14] Halmos, P. R., “Measure Theory,” Princeton, New Jersey:D. Van Nostrand Co., Inc., 1950. [15] Kelley, J. L., “General Topology,” Princeton, New Jersey:D. Van Nostrand Co., Inc., 1955. Even this always does not yield uniqueness, but we will show the additional restriction that will guarantee uniqueness after the necessary language is developed. Since all metrizations of a given metrizable topology are isomorphic, in the quotient class the orthogonal Euclidean geometry serves the purpose of being a convenient representative of the unique element resulting from a given metrizable topology. Furthermore, the same comment applies to the use of a Gaussian distribution as the probability distribution on this orthogonal Euclidean geometry. Namely, the random Gaussian distribution on an orthogonal Euclidean geometry is a convenient representative member of the equivalence class which maps into one element (stochastic space) of the quotient class. Information Theory