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Question 1 Let M. be a compact set. Then. (A) M must be infinite (B) If (Fn) is a decreasing sequence of nonempty closed subsets of M; then Fn = 0 nEN* (C) Every closed subset of M must be bounded (D) M must be finite.

Question 1 Let M. be a compact set. Then. (A) M must be infinite (B) If (Fn) is a decreasing sequence of nonempty closed subsets of M; then Fn = 0 nEN* (C) Every closed subset of M must be bounded (D) M must be finite.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 13E: 13. Consider the set of all nonempty subsets of . Determine whether the given relation on is...
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Part 1: Multiple Choice Question 1 Let. M. be a compact set. Then (A) M must be infinite (B) If (Fn) is a decreasing sequence of nonempty closed subsets of M; then Fn = 0 nEN* (C) Every closed subset of M must be bounded (D) M must be finite. Question 3 Let c E M. Which of the following is false? A) The function f: M → M: x + c = f(x) is false? Lipschitz B) The function.f: M→M: x x is uniformly continuous C) every k-Lipschitz map is a contraction map P) Every Lipschitz function is continuous S Question 2 rks Let M be a metric space. Which of the following is false? (A). Every finite subset of M is compact (B) Every bounded sequence is conver- gent (C) For all x, y € M, there exists some r>0 such that y = B(x,r) (D) A finite union of compact subsets of M is compact Question 4 Let f: MM be a uniformly continu- ous function. Which of the following is (A) If A C M, then f-¹(A) must be ei- ther open or closed. (B) If E and B are open subsets of M with ECB, then f−¹(Eª U Bº) is a. closed subset of (f-¹(E))º` (C) V > 0, 36> 0 such that Vx, y ≤ M, d(x, y)<8⇒d(f(x), f(y)) < e. (D) For each co E M, and Ve > 0, 3 such that Vx € M, d(x,xo) < S d(f(x), f(xo)) < e.
Transcribed Image Text:Part 1: Multiple Choice Question 1 Let. M. be a compact set. Then (A) M must be infinite (B) If (Fn) is a decreasing sequence of nonempty closed subsets of M; then Fn = 0 nEN* (C) Every closed subset of M must be bounded (D) M must be finite. Question 3 Let c E M. Which of the following is false? A) The function f: M → M: x + c = f(x) is false? Lipschitz B) The function.f: M→M: x x is uniformly continuous C) every k-Lipschitz map is a contraction map P) Every Lipschitz function is continuous S Question 2 rks Let M be a metric space. Which of the following is false? (A). Every finite subset of M is compact (B) Every bounded sequence is conver- gent (C) For all x, y € M, there exists some r>0 such that y = B(x,r) (D) A finite union of compact subsets of M is compact Question 4 Let f: MM be a uniformly continu- ous function. Which of the following is (A) If A C M, then f-¹(A) must be ei- ther open or closed. (B) If E and B are open subsets of M with ECB, then f−¹(Eª U Bº) is a. closed subset of (f-¹(E))º` (C) V > 0, 36> 0 such that Vx, y ≤ M, d(x, y)<8⇒d(f(x), f(y)) < e. (D) For each co E M, and Ve > 0, 3 such that Vx € M, d(x,xo) < S d(f(x), f(xo)) < e.
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