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Math 321 Real Analysis
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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Math 321 Real Analysis
PopularMetric 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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Examen
Math 321 Real Analysis
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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Math 321 Real Analysis
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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Examen
Math 321 Real Analysis
Math 321 Real Analysis David Perkinson Contents Foreword
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