5. State whether each of the following statements is true or false, in either case substantiate/ justify. (a) Let f: Q→Q be a contraction on Q. Then by Banach' s fixed point theorem there is a unique point x EQ such that f(x) = x. (b) Let X = {x €R:-1
5. State whether each of the following statements is true or false, in either case substantiate/ justify. (a) Let f: Q→Q be a contraction on Q. Then by Banach' s fixed point theorem there is a unique point x EQ such that f(x) = x. (b) Let X = {x €R:-1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![5. State whether each of the following statements is true or false, in either case substantiate/
justify.
(a) Let f: Q→Q be a contraction on Q. Then by Banach' s fixed point theorem there is a
unique point x EQ such that f(x) = x.
(b) Let X = {x €R :-1<xs 0 or 0 < x s 1} with the usual metric. Then X is disconnected.
(c) The set X = [3,9] as a subset of R with the discrete metric is disconnected.[03]
(d) Every continuous map on any metric space is an open map.
(e) Every connected metric spaces is compact.
(f) The set B = {(xx) ER2 : x² + y2 < 1} as a subset of R2 with indiscrete metric is
disconnected.
(g) The empty set is path-connected.
(h) The set of natural numbers as a subset of real numbers with the usual metric is
compact.
(i) Let fand g be uniformly continuous on a metric space X into R. Then the pr
g is uniformly continuous on X into R. |](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31177463-1db3-4cd5-81cb-4b87f5733f5d%2F221533f9-0ef4-4e60-af9e-e93f65160def%2Fipus7u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5. State whether each of the following statements is true or false, in either case substantiate/
justify.
(a) Let f: Q→Q be a contraction on Q. Then by Banach' s fixed point theorem there is a
unique point x EQ such that f(x) = x.
(b) Let X = {x €R :-1<xs 0 or 0 < x s 1} with the usual metric. Then X is disconnected.
(c) The set X = [3,9] as a subset of R with the discrete metric is disconnected.[03]
(d) Every continuous map on any metric space is an open map.
(e) Every connected metric spaces is compact.
(f) The set B = {(xx) ER2 : x² + y2 < 1} as a subset of R2 with indiscrete metric is
disconnected.
(g) The empty set is path-connected.
(h) The set of natural numbers as a subset of real numbers with the usual metric is
compact.
(i) Let fand g be uniformly continuous on a metric space X into R. Then the pr
g is uniformly continuous on X into R. |
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