Metric Spaces 1

The articles [9], [5], [11], [1], [10], [6], [3], [4], [2], [8], [12], and [7] provide the notation and terminology for this paper. We introduce metric structures which are extensions of 1-sorted structure and are systems 〈 a carrier, a distance 〉, where the carrier is a set and the distance is a function from [: the carrier, the carrier :] into R. One can check that there exists a metric structure which is non empty and strict. Let A, B be sets, let f be a partial function from [:A, B :] to R, let a be an element of A, and let b be an element of B. Then f (a, b) is a real number. Let M be a metric structure and let a, b be elements of M. The functor ρ(a,b) yields a real number and is defined by: