10. Let X be the set of real numbers, and let d be the pseudometric given by d(x, y) = 1 ifry, and d(x,x) = 0 (Example 1.1(e)). (a) Describe the closure of the cell C(0; 1). (b) Describe the set B(0:1) = {y € X: d(0, y) ≤ 1}.
10. Let X be the set of real numbers, and let d be the pseudometric given by d(x, y) = 1 ifry, and d(x,x) = 0 (Example 1.1(e)). (a) Describe the closure of the cell C(0; 1). (b) Describe the set B(0:1) = {y € X: d(0, y) ≤ 1}.
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
Basic Pseudometric space: For the question 10a, the solution was provided in 2nd photo, can you look over it and produce the graph of the set described?
![30 Chapter 1 Pseudometric Spaces
Exercises
10. Let X be the set of real numbers, and let d be the pseudometric given by
d(x, y) = 1 if z #y, and d(x,x) = 0 (Example 1.1(e)).
(a) Describe the closure of the cell C(0; 1).
(b) Describe the set B(0:1) = {y E X: d(0, y) ≤ 1}.
11. Let (X. d) be a pseudometric space, and suppose d has the property that
d(a, b) > 0 whenever a b. Prove that every finite subset of X is closed.
12. Let (X, d) be the space of real numbers with the usual pseudometric. For each
positive integer n, let A₁ = (1/n, 1]. Find cl(U{A}), and find U{cl(An)}.
An](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9787c00-cfcc-43da-9235-b2564f9dd12e%2F53665c65-1a74-4753-be62-6ee6cde911af%2Fm02o715_processed.jpeg&w=3840&q=75)
Transcribed Image Text:30 Chapter 1 Pseudometric Spaces
Exercises
10. Let X be the set of real numbers, and let d be the pseudometric given by
d(x, y) = 1 if z #y, and d(x,x) = 0 (Example 1.1(e)).
(a) Describe the closure of the cell C(0; 1).
(b) Describe the set B(0:1) = {y E X: d(0, y) ≤ 1}.
11. Let (X. d) be a pseudometric space, and suppose d has the property that
d(a, b) > 0 whenever a b. Prove that every finite subset of X is closed.
12. Let (X, d) be the space of real numbers with the usual pseudometric. For each
positive integer n, let A₁ = (1/n, 1]. Find cl(U{A}), and find U{cl(An)}.
An
![n/collab/ui/session/playback
1,2
#10
X = R
1 if x #y
d (x,y) = { 0 if x = y
@ Find cl (C(0; 1),
디
{0}
C (0:1) = {x: d(0₁ x) < 1} = {0}
C(0; 2) = { x: d (0₁ x) <2}
=IR
pecl S every C(pir) meets S Prop. 1.4
D
C (0;r)
O Ed S
every
meets S
Decl ({0})+ every ((0;r) meet {0}
either {0} or IR
1
1
if r²1
Claim cl (C(0;11) = {0]. What if p=0.1 Ecl (5) = cl ({0})?
Then every ((0.1;r) meets {0}, which is falte.
Note ((0, 1) { x: 2 (0.1₁x) <1} = {0.1}
if r>l
53:01
⇒
1x
li
CC](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9787c00-cfcc-43da-9235-b2564f9dd12e%2F53665c65-1a74-4753-be62-6ee6cde911af%2Fuswf694_processed.jpeg&w=3840&q=75)
Transcribed Image Text:n/collab/ui/session/playback
1,2
#10
X = R
1 if x #y
d (x,y) = { 0 if x = y
@ Find cl (C(0; 1),
디
{0}
C (0:1) = {x: d(0₁ x) < 1} = {0}
C(0; 2) = { x: d (0₁ x) <2}
=IR
pecl S every C(pir) meets S Prop. 1.4
D
C (0;r)
O Ed S
every
meets S
Decl ({0})+ every ((0;r) meet {0}
either {0} or IR
1
1
if r²1
Claim cl (C(0;11) = {0]. What if p=0.1 Ecl (5) = cl ({0})?
Then every ((0.1;r) meets {0}, which is falte.
Note ((0, 1) { x: 2 (0.1₁x) <1} = {0.1}
if r>l
53:01
⇒
1x
li
CC
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