n) The set of natural numbers as a subset of real numbers with the usual metric is compact.
n) The set of natural numbers as a subset of real numbers with the usual metric is compact.
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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Question
Help me with no (h)
![5. State whether each of the following statements is true or false, in either case substan-
tiate/ justify.
(a) Let f : Q + Q be a contraction on Q. Then by Banach' s fixed point theorem
there is a unique point z € Q such that f(x) = x.
(b) Let X = {x € R : -1 < 1 < 0 or 0 <I< 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
(d) Every continuous map on any metric space is an open map.
(e) Every connected metric spaces is compact.
(f) The set B = {(z, y) E R² : z² + 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.
(1) Let f and g be uniformly continuous on a metric space X into R. Then the
product f * g is uniformly continuous on X into R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e8d32e9-8105-46ce-9057-29bc969fd459%2F46e353af-6bae-4c5d-9c4b-2c7a91422f70%2Ffaofsj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5. State whether each of the following statements is true or false, in either case substan-
tiate/ justify.
(a) Let f : Q + Q be a contraction on Q. Then by Banach' s fixed point theorem
there is a unique point z € Q such that f(x) = x.
(b) Let X = {x € R : -1 < 1 < 0 or 0 <I< 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
(d) Every continuous map on any metric space is an open map.
(e) Every connected metric spaces is compact.
(f) The set B = {(z, y) E R² : z² + 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.
(1) Let f and g be uniformly continuous on a metric space X into R. Then the
product f * g is uniformly continuous on X into R.
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