Totally Bounded Metric Spaces

Abstract

The papers [19], [9], [1], [4], [20], [2], [18], [13], [5], [8], [14], [21], [7], [15], [12], [11], [17], [6], [10], [16], and [3] provide the terminology and notation for this paper. For simplicity we follow the rules: M is a metric space, c, g are elements of the carrier of M , F is a family of subsets of the carrier of M , A, B are subsets of the carrier of M , f is a function, n, m, p, k are natural numbers, and r, s, L are real numbers. Next we state four propositions: (1) For every L such that 0 0 there exists F such that F is finite and the carrier of M = ⋃ F and for every A such that A ∈ F there exists g such that A = Ball(g, r).