Let X d Be a Compact Metric Space Let C X Denote the Continuous Functions From X to C
Transcribed Text
Q2. Let (X, d) be a metric space with A C X and x € X. Let A denote the closure of A in X. (a) (i) Determine (without proof) the closure of Z in R. [3 marks] (ii) Show that x € A if and only if Gn A # 0 for every open subset G of X containing x. 5 marks] (b) Exhibit three nonempty proper subsets B, C, D of R (with its usual Euclidean metric) such that (i) B contains at least one accumulation point of B and at least one point that is not an accumulation point of B, 2 marks] (ii) C contains no accumulation points of C, [2 marks] (iii) all elements of D are accumulation points of D. 2 marks] (c) Indicate whether each of the following statements is true or false, providing an explanation or counterexample as appropriate in each case: (i) If x is an accumulation point of A, then x € A. [2 marks] (ii) If x is an accumulation point of A, then x E A. 2 marks] (iii) If A is a closed set and x is an accumulation point of A, then x € A. 2 marks] (d) Determine with proof a connection between A and the set of interior points of X \ A. 5 marks] Q3. Let X and Y be metric spaces with A C X. (a) (i) Let f : R² R be defined by f fxxy = y. Show, by preservation of sequential convergence or otherwise, that f is continuous. 4 marks] (ii) Complete the following statement: A is closed in X if and only if for each convergent sequence (an) of elements from A, 2 marks] (iii) Show that if A is closed and (an) is a sequence in A converging to l, then l € A. [2 marks] (b) By considering part (a) or otherwise, show that S = {(x,y) € R2: y = x # 0} is closed in R². [6 marks] (c) Show that if g and h are continuous mappings of X to Y such that g(a) = h(a) for every a € A, it must follow that g(x) = h(x) for every x € A. [6 marks] (d) It is known that the metric space C[0, 1], the set of all continuous real-valued functions on [0, 1] equipped with the uniform metric du, is complete. Sketch a brief outline of how one may deduce this fact. [5 marks] Q4. (a) (i) Give an example of an infinite compact subset of IR that is not an interval. 2 marks] (ii) Show why R is not compact. 2 marks (iii) Prove that every compact subspace of a metric space is bounded. 3 marks] (b) Let f : X Y be a continuous onto function where X and Y are metric spaces. (i) Show that if X is compact, then Y must also be compact. [4 marks] (ii) Hence explain why no continuous function from the interval [0, 1] onto R can exist. [3 marks] (iii) By considering the function g(x) = tan - 1 (x) or otherwise, show that if X is complete, Y need not be complete. [4 marks] (c) Explain carefully why C[0,1] with the mean metric ds is not complete. [7 marks]
These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.
By purchasing this solution you'll be able to access the following files:
Solution.pdf.
Source: https://www.24houranswers.com/college-homework-library/Mathematics/Real-Analysis/64437
0 Response to "Let X d Be a Compact Metric Space Let C X Denote the Continuous Functions From X to C"
Post a Comment