Vol.2, No.4, September–October 1991 Université Catholique de Louvain Totally Bounded Metric Spaces MML Identifier: TBSP 1. Alicia

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 m ≤ L n. (2) For every L such that 0 < L and L < 1 for every k holds L k ≤ 1 and 0 < L k. (3) For every L such that 0 < L and L < 1 for every s such that 0 < s there exists n such that L n < s. (4) For every set X such that X is finite and X � = ∅ 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). Let us consider M. A function is called a sequence of M if: (Def.2) dom it = ¢ and rng it ⊆ the carrier of M. In the sequel S1 will denote a sequence of M. The following proposition is tru