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Metric Spaces 1. Equivalence Relations Definition 1.1. A relation on a set S is a subset R ⊆ S × S. If (a, b) ∈ R, we write a ∼R b (or just a ∼ b). Example 1.2. (1) Let S be the set of members of our class, and let (a, b) ∈ R if person a is shorter than person b. (2) Let S = Z and R = {(a, b) ∈ Z | b = 2a}. Thus, 4 ∼ 8 and 8 ∼ 16. Definition 1.3. A relation ∼ on a set S is an equivalence relation if for all a, b, c ∈ S, (1) a ∼ a (reflexivity) (2) a ∼ b =⇒ b ...

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81 pages

Written in 2023/2024

Grade A+

MYSOLUTIONS

Chapter 1 Metric Spaces 1. Equivalence Relations Denition 1.1. A relation on a set S is a subset R  S S. If (a; b) 2 R, we write a R b (or just a  b). Example 1.2. (1) Let S be the set of members of our class, and let (a; b) 2 R if person a is shorter than person b. (2) Let S = Z and R = f(a; b) 2 Z j b = 2ag. Thus, 4  8 and 8  16. Denition 1.3. A relation  on a set S is an equivalence relation if for all a; b; c 2 S, (1) a  a (re exivity) (2) a  b =) b  a (symmetry) (3)...

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81 pages

Written in 2021/2022

StudyHubSolutions

Metric Spaces 1. Equivalence Relations Definition 1.1. A relation on a set S is a subset R ⊆ S × S. If (a, b) ∈ R, we write a ∼R b (or just a ∼ b). Example 1.2. (1) Let S be the set of members of our class, and let (a, b) ∈ R if person a is shorter than person b. (2) Let S = Z and R = {(a, b) ∈ Z | b = 2a}. Thus, 4 ∼ 8 and 8 ∼ 16. Definition 1.3. A relation ∼ on a set S is an equivalence relation if for all a, b, c ∈ S, (1) a ∼ a (reflexivity) (2) a ∼ b =⇒ b ...

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81 pages

Written in 2023/2024

Grade A+

TheInstructor

Math 321 Real Analysis David PerkinsonContents Foreword Chapter 1. i Metric Spaces1 §1.Equivalence Relations1 §2.Cardinality I2 §3.Cardinality II3 §4.Metric Spaces4 §5.Closed Sets and Limits5 §6.Continuity7 §7.Miscellaneous8 §8.Completeness9 §9.Completeness II11 §10.Picard’s Theorem12 Chapter 2. Topological Spaces 15 §1.Topology15 §2.Separation Axioms16 §3.Connectedness17 §4.Compactness I18 §5.Compactness In Metric Spaces19 §6.Compactness in Metric Spac...

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81 pages

Written in 2022/2023

BESTEXAMINER01

Math 321 Real Analysis David Perkinson Contents Foreword

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81 pages

Written in 2021/2022

Grade A

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