Corollary 2.54 Let (M, d) be a complete metric space and K C M. The following statements are equivalent: (i) K is relatively compact. (іi) К is totally bounded.
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- Which characteristics implies(y) that a set S of real numbers is closed? a. S has no accumulation points b. S is bounded c. S has an open cover that contains a finite subcover d. None of the above16. The set S = { x∈R: x2 - 4<0} with the usual metric is .......................... A. Compact. B. Connected. C. Not connected. D. Sequentially compact.d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. Conclude that, with this metric, a subset of ℝ3 is sequentially compact if and only if it is closed and bounded.
- Let (M, d) be a metric space and K ⊂ M. The following statements are equivalent: (i) K is relatively compact. (ii) K is totally bounded and (K^_, d) is complete. 2. Let (M, d) be a metric space and K ⊂ M. The following statements are equivalent: (i) K is relatively compact.(ii) K is totally bounded.Prove Corollary 5.5.10 - Let f be a continuous real-valued function defined on a metric space (X, d), and let D be a compact subset of X. Then f assumes maximum and minimum values on D.Is the metric space (L, dL ) sequentially compact? Justify your answer.