Totally Bounded Metric Spaces

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 < L and L < 1 for all n, m such that n ≤ m holds L ≤ L. (2) For every L such that 0 < L and L < 1 for every k holds L ≤ 1 and 0 < L. (3) For every L such that 0 < L and L < 1 for every s such that 0 < s there exists n such that L < s. (4) For every set X such that X is finite and X 6= ∅ and for all sets Y , Z such that Y ∈ X and Z ∈ X holds Y ⊆ Z or Z ⊆ Y there exists a set V such that V ∈ X and for every set Z such that Z ∈ X holds V ⊆ Z. Let us consider M , F . Then ⋃ F is a subset of the carrier of M . Let D be a non-empty set. Then ΩD is a subset of D. Then ∅D is a subset of D. Let us consider M . We say that M is totally bounded if and only if: (Def.1) for every r such that r > 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).