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Exercise 1. 1) In a metric space (X, d), show that any closed ball B(x,r), with x X and r≥0 is closed subset. 2) Give an example of a metric space in which no closed ball is open. 3) Give an example of a space in which all closed balls are open. 4) In the plane R², for v € R²\{0}, consider the open half line D, through {0} defined by D₂ = {tv | t >0} CR². 0 & Dv Show that D, is not a closed subset of R2. Show that D₁ U {0} is a closed subset in R². Dv 5) Consider Un = (1, 1/n) v = (1,0) in R2. Show that neither Unzi Dvn nor {0} UUn≥1Dvn is a closed subset of R². Dy Dv₂.

Exercise 1. 1) In a metric space (X, d), show that any closed ball B(x,r), with x X and r≥0 is closed subset. 2) Give an example of a metric space in which no closed ball is open. 3) Give an example of a space in which all closed balls are open. 4) In the plane R², for v € R²\{0}, consider the open half line D, through {0} defined by D₂ = {tv | t >0} CR². 0 & Dv Show that D, is not a closed subset of R2. Show that D₁ U {0} is a closed subset in R². Dv 5) Consider Un = (1, 1/n) v = (1,0) in R2. Show that neither Unzi Dvn nor {0} UUn≥1Dvn is a closed subset of R². Dy Dv₂.

Advanced Engineering Mathematics
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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Need Parts 1, 2, 3, and 4

Exercise 1. 1) In a metric space (X, d), show that any closed ball B(x,r), with x X and r≥0 is closed subset. 2) Give an example of a metric space in which no closed ball is open. 3) Give an example of a space in which all closed balls are open. 4) In the plane R², for v € R²\{0}, consider the open half line D, through {0} defined by D₂ = {tv | t >0} CR². 0 & Dv Show that D, is not a closed subset of R2. Show that D, U {0} is a closed subset in R². Dv 5) Consider Un = (1, 1/n) v = (1,0) in R2. Show that neither Unzi Dvn nor {0} UUn≥1Dvn is a closed subset of R². DVI Dv₂, Dup
Transcribed Image Text:Exercise 1. 1) In a metric space (X, d), show that any closed ball B(x,r), with x X and r≥0 is closed subset. 2) Give an example of a metric space in which no closed ball is open. 3) Give an example of a space in which all closed balls are open. 4) In the plane R², for v € R²\{0}, consider the open half line D, through {0} defined by D₂ = {tv | t >0} CR². 0 & Dv Show that D, is not a closed subset of R2. Show that D, U {0} is a closed subset in R². Dv 5) Consider Un = (1, 1/n) v = (1,0) in R2. Show that neither Unzi Dvn nor {0} UUn≥1Dvn is a closed subset of R². DVI Dv₂, Dup
Show that D, is not a closed subset of R2. Show that DU {0} is a closed subset in R². 5) Consider Un = (1,1/n) → v∞ = (1,0) in R2. Show that neither Unzi Dvn nor {0} UUn>1Dvn is a closed subset of R². V1 DVI Dv₂ 1 ****** ******* Dv₂ Dvo Show that {0} U Dvx U (Un>1Dvn) is a closed subset of R². Can you find a sequence v₁ € R², n ≥ 1 such that vn → V∞ ‡ 0 for which {0} UUn>1Dvn is a closed subset?
Transcribed Image Text:Show that D, is not a closed subset of R2. Show that DU {0} is a closed subset in R². 5) Consider Un = (1,1/n) → v∞ = (1,0) in R2. Show that neither Unzi Dvn nor {0} UUn>1Dvn is a closed subset of R². V1 DVI Dv₂ 1 ****** ******* Dv₂ Dvo Show that {0} U Dvx U (Un>1Dvn) is a closed subset of R². Can you find a sequence v₁ € R², n ≥ 1 such that vn → V∞ ‡ 0 for which {0} UUn>1Dvn is a closed subset?
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