MML Identifier: TBSP_1. Totally Bounded Metric Spaces Alicia de la Cruz

and [14] provide the notation and terminology for this paper. For simplicity, we follow the rules: M denotes a non empty metric space, c denotes an element of M, N denotes a non empty metric structure, w denotes an element of N, G denotes a family of subsets of N, C denotes a subset of N, R denotes a Reflexive non empty metric structure, T denotes a Reflexive symmetric triangle non empty metric structure, t1 denotes an element of T, Y denotes a family of subsets of T, f denotes a function, n, m, p, k denote natural numbers, r, s, L denote real numbers, and x denotes a set. Next we state three propositions: (1) For every L such that 0 < L and L < 1 and for all n, m such that n ≤ m holds L m ≤ L n. (2) For every L such that 0 < L and L < 1 and for every k holds L k ≤ 1 and 0 < L k. (3) For every L such that 0 < L and L < 1 and for every s such that 0 < s there exists n such that L n < s. Let us consider N. We say that N is totally bounded if and only if the condition (Def. 1) is satisfied. (Def. 1) Let given r. Suppose r> 0. Then there exists G such that G is finite and the carrier of N = � G and for every C such that C ∈ G there exists w such that C = Ball(w,r). Let us consider N. We see that the sequence of N is a function and it can be characterized by the following (equivalent) condition: (Def. 2) domit = N and rngit ⊆ the carrier of N. In the sequel S1 is a sequence of M and S2 is a sequence of N. We now state the proposition (5) 1 f is a sequence of N iff dom f = N and for every n holds f (n) is an element of N. Let us consider N, S2. We say that S2 is convergent if and only if: (Def. 3) There exists an element x of N such that for every r such that r> 0 there exists n such that for every m such that n ≤ m holds ρ(S2(m),x) < r. Let us consider M, S1. Let us assume that S1 is convergent. The functor limS1 yields an element of M and is defined by: 1 The proposition (4) has been removed