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(g) The empty set is path-connected. (h) The set of natural numbers as a subs

(g) The empty set is path-connected. (h) The set of natural numbers as a subs

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 30EQ
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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.
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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