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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
#1
Need Parts 1, 2, 3, and 4
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
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?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
