An Introduction to Metric
Spaces 1st Edition
ST
UV
SOLUTIONS
IA
_A
MANUAL
PP
Dhananjay Gopal
RO
Aniruddha Deshmukh
VE
Comprehensive Solutions Manual for Instructors and
Students
D?
© Dhananjay Gopal & Aniruddha Deshmukh
?
All rights reserved. Reproduction or distribution without permission is prohibited.
© DREAMSHUB
, Solutions Manual Companion for An Introduction to Metric Spaces (1st
Edition)
Dhananjay Gopal & Aniruddha Deshmukh
ISBN: 9780367493493
ST
UNIT 1: FOUNDATIONS OF MATHEMATICAL STRUCTURES
1. Set Theory
UV
UNIT 2: METRIC SPACE FRAMEWORKS
2. Metric Spaces
3. Complete Metric Spaces
UNIT 3: TOPOLOGICAL PROPERTIES OF METRIC SPACES
IA
4. Compact Metric Spaces
5. Connected Spaces
_A
UNIT 4: CONTINUITY AND MAPPINGS
6. Continuity
UNIT 5: FIXED POINT THEORY AND APPLICATIONS
7. Banach Fixed Point Theorem and Its Applications
PP
RO
VE
D?
?
© DREAMSHUB
, Solutions Manual for An Introduction to Metric
Spaces, 1e Dhananjay Gopal, Aniruddha
Deshmukh (All Chapters)
Chapter 1
Set Theory
ST
Ex 1.1. Collection of lines with constant slope and changing intercepts (or, vice-versa) does the job.
That is,
UV
F = {Lr |r ∈ R} ,
where, Lr = {(x, y) ∈ R2 |x + y = r} are lines with slope −1 and intercept r. Other similar sets can be
constructed which can be indexed by R.
Ex 1.2. A few subsethood relations are
IA
N⊆Z⊆Q⊆R
N ⊆ Q+ ⊆ R+ ⊆ R
N ⊆ Q∗ ⊆ R∗ ⊆ R
_A
Similar relations can be obtained by comparing elements.
Ex 1.3. The results for N are given. Other results can be constructed in a similar manner.
PP
N∪Z=Z
N∩Z=N
N∪Q=Q
N∩Q=N
RO
N ∪ Q+ = Q+
N ∩ Q+ = N
N ∪ Q− = {x ∈ Q|x < 0 or x ∈ N}
N ∩ Q− = ∅
N ∪ Q∗ = Q∗
VE
N ∩ Q∗ = N
N ∪ R + = R+
N ∩ R+ = N
N ∪ R− = {x ∈ R|x < 0 or x ∈ N}
D?
N ∩ R− = ∅
N ∪ R∗ = R∗
N ∩ R∗ = N
?
A few observations that can be made are as follows. The reader can make as many observations as they
want to.
1. A ⊂ B implies A ∪ B = B and A ∩ B = A.
2. Some of the (newly formed) sets are merely a result of union/intersection of known sets.
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Spaces 1st Edition
ST
UV
SOLUTIONS
IA
_A
MANUAL
PP
Dhananjay Gopal
RO
Aniruddha Deshmukh
VE
Comprehensive Solutions Manual for Instructors and
Students
D?
© Dhananjay Gopal & Aniruddha Deshmukh
?
All rights reserved. Reproduction or distribution without permission is prohibited.
© DREAMSHUB
, Solutions Manual Companion for An Introduction to Metric Spaces (1st
Edition)
Dhananjay Gopal & Aniruddha Deshmukh
ISBN: 9780367493493
ST
UNIT 1: FOUNDATIONS OF MATHEMATICAL STRUCTURES
1. Set Theory
UV
UNIT 2: METRIC SPACE FRAMEWORKS
2. Metric Spaces
3. Complete Metric Spaces
UNIT 3: TOPOLOGICAL PROPERTIES OF METRIC SPACES
IA
4. Compact Metric Spaces
5. Connected Spaces
_A
UNIT 4: CONTINUITY AND MAPPINGS
6. Continuity
UNIT 5: FIXED POINT THEORY AND APPLICATIONS
7. Banach Fixed Point Theorem and Its Applications
PP
RO
VE
D?
?
© DREAMSHUB
, Solutions Manual for An Introduction to Metric
Spaces, 1e Dhananjay Gopal, Aniruddha
Deshmukh (All Chapters)
Chapter 1
Set Theory
ST
Ex 1.1. Collection of lines with constant slope and changing intercepts (or, vice-versa) does the job.
That is,
UV
F = {Lr |r ∈ R} ,
where, Lr = {(x, y) ∈ R2 |x + y = r} are lines with slope −1 and intercept r. Other similar sets can be
constructed which can be indexed by R.
Ex 1.2. A few subsethood relations are
IA
N⊆Z⊆Q⊆R
N ⊆ Q+ ⊆ R+ ⊆ R
N ⊆ Q∗ ⊆ R∗ ⊆ R
_A
Similar relations can be obtained by comparing elements.
Ex 1.3. The results for N are given. Other results can be constructed in a similar manner.
PP
N∪Z=Z
N∩Z=N
N∪Q=Q
N∩Q=N
RO
N ∪ Q+ = Q+
N ∩ Q+ = N
N ∪ Q− = {x ∈ Q|x < 0 or x ∈ N}
N ∩ Q− = ∅
N ∪ Q∗ = Q∗
VE
N ∩ Q∗ = N
N ∪ R + = R+
N ∩ R+ = N
N ∪ R− = {x ∈ R|x < 0 or x ∈ N}
D?
N ∩ R− = ∅
N ∪ R∗ = R∗
N ∩ R∗ = N
?
A few observations that can be made are as follows. The reader can make as many observations as they
want to.
1. A ⊂ B implies A ∪ B = B and A ∩ B = A.
2. Some of the (newly formed) sets are merely a result of union/intersection of known sets.
Downloaded by: tutorsection | Want to earn $1.236
Distribution of this document is illegal extra per year?
1
