Difference between revisions of "MAT3333"

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'''Catalog entry:'''
 
'''Catalog entry:'''
−
MAT 333 Fundamentals of Analysis and Topology.
+
MAT 3333 Fundamentals of Analysis and Topology.
 
Prerequisite: MAT 3003 Discrete Mathematics, or consent of instructor.
 
Prerequisite: MAT 3003 Discrete Mathematics, or consent of instructor.
−
Topology of the real line. Introduction to point-set topology.
+
Topological notions in the real line and in metric spaces. Convergent sequences. Continuous functions. Connected and compact sets. The Intermediate Value and Extreme Value theorems. Sequential compactness and the Heine-Borel Theorem.
  
 
'''Prerequisites:'''
 
'''Prerequisites:'''
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|
 
<!-- Week # -->
 
<!-- Week # -->
−
1
+
1-3
 
||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
1.1. Appendices C, G & H.
+
Chapters 1 & 2. Appendices C, G & H.
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
 
Operations, order and intervals of the real line.
 
Operations, order and intervals of the real line.
 +
Completeness of the real line. Suprema and infima.
 +
Basic topological notions in the real line.
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
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* Field axioms.
 
* Field axioms.
 
* Order of ℝ.
 
* Order of ℝ.
−
 
−
|-
 
−
|
 
−
<!-- Week # -->
 
−
2
 
−
||
 
−
<!-- Sections -->
 
−
1.2. Appendix J.
 
−
||
 
−
<!-- Topics -->
 
−
Completeness of the real line. Suprema and infima.
 
−
||
 
−
<!-- SLOs -->
 
 
* Intervals: open, closed, bounded and unbounded.
 
* Intervals: open, closed, bounded and unbounded.
 
* Upper and lower bounds of subsets of ℝ.
 
* Upper and lower bounds of subsets of ℝ.
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* The Least Upper Bound Axiom (completeness of ℝ).
 
* The Least Upper Bound Axiom (completeness of ℝ).
 
* The Archimedean property of ℝ.
 
* The Archimedean property of ℝ.
−
 
−
|-
 
−
|
 
−
3
 
−
||
 
−
2.1, 2.2
 
−
||
 
−
Basic topological notions in the real line.
 
−
||
 
 
* Distance.
 
* Distance.
 
* Neighborhoods and interior of a set.
 
* Neighborhoods and interior of a set.
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||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
3.1–3.3
+
Chapter 3
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
−
Continuous functions on subsets of the real line.
+
Continuous functions on ℝ.
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
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|
 
<!-- Week # -->
 
<!-- Week # -->
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5
+
4-5
 
||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
---
+
Chapter 4
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
−
Review. First midterm exam.
+
Convergence of real sequences.
 +
The Cauchy criterion. Subsequences.
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
 +
* Sequences in ℝ.
 +
* Convergent sequences.
 +
* Algebraic operations on convergent sequences.
 +
* Sufficient conditions for convergence. Cauchy criterion.
 +
* Subsequences.
  
 
|-
 
|-
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||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
4.1, 4.2
+
Chapter 5
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
−
Convergence of real sequences.
+
Connectedness and the Intermediate Value Theorem
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
−
* Sequences in ℝ.
+
* Connected subsets of ℝ.
−
* Convergent sequences.
+
* Continuous images of connected sets.
−
* Algebraic operations on convergent sequences.
+
* The Intermediate Value Theorem.
  
 
|-
 
|-
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||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
4.3, 4.4
+
Chapter 6
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
−
The Cauchy criterion. Subsequences.
+
Compactness and the Extreme Value Theorem.
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
−
* Sufficient conditions for convergence. Cauchy criterion.
+
* Compact subsets of the real line.
−
* Subsequences.
+
* Examples of compact subsets.
 +
* The Extreme Value Theorem.
  
 
|-
 
|-
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||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
5.1, 5.2
+
Chapter 7
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
−
Connectedness and the Intermediate Value Theorem
+
Limits of real functions.
 +
||
 +
<!-- SLOs -->
 +
* Limit of a real function at a point.
 +
* Continuity and limits.
 +
* Arithmetic properties of limits.
 +
 
 +
|-
 +
|
 +
<!-- Week # -->
 +
9 & 10
 +
||
 +
<!-- Sections -->
 +
Chapters 9, 10, 11
 +
||
 +
<!-- Topics -->
 +
The topology of metric spaces.
 +
||
 +
<!-- SLOs -->
 +
* Metric spaces. Examples.
 +
* Equivalent metrics.
 +
* Interior, closure, and boundary.
 +
* Accumulation point.
 +
* Boundary point.
 +
* Closure.
 +
* Open and closed sets.
 +
* The relative topology.
 +
 
 +
|-
 +
|
 +
<!-- Week # -->
 +
11
 +
||
 +
<!-- Sections -->
 +
Chapter 12
 +
||
 +
<!-- Topics -->
 +
Sequences in metric spaces.
 +
||
 +
<!-- SLOs -->
 +
* Convergent sequences.
 +
* Sequential characterizations of topological properties.
 +
 
 +
|-
 +
|
 +
<!-- Week # -->
 +
12
 +
||
 +
<!-- Sections -->
 +
Chapter 14
 +
||
 +
<!-- Topics -->
 +
Continuity and limits.
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
−
* Connected subsets of ℝ.
+
* Continuous functions between metric spaces.
−
* Continuous images of connected sets.
+
* Topological products.
−
* The Intermediate Value Theorem.
+
* Limits.
  
 
|-
 
|-
 
|
 
|
 
<!-- Week # -->
 
<!-- Week # -->
−
9
+
13
 
||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
6.1, 6.2, 6.3
+
15.1–15.2
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
−
Compactness and the Extreme Value Theorem.
+
Compact metric spaces.
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
−
* Compact subsets of the real line.
+
* Compactness: definition and elementary properties.
−
* Examples of compact subsets.
 
 
* The Extreme Value Theorem.
 
* The Extreme Value Theorem.
  
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|
 
|
 
<!-- Week # -->
 
<!-- Week # -->
−
10
+
14
 
||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
7.1, 7.2
+
Chapter 16
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
−
Limits of real functions.
+
Sequential compactness and the Heine-Borel Theorem.
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
−
* Limit of a real function at a point.
+
* Sequential compactness.
−
* Continuity and limits.
+
* Conditions equivalent to compactness of a metric space.
−
* Arithmetic properties of limits.
+
* The Heine-Borel Theorem.
  
 +
|-
 +
|
 +
<!-- Week # -->
 +
15
 +
||
 +
<!-- Sections -->
 +
Chapter 18 (time permitting)
 +
||
 +
<!-- Topics -->
 +
Complete metric spaces.
 +
||
 +
<!-- SLOs -->
 +
* Cauchy sequences in metric spaces.
 +
* Metric completness.
 +
* Completeness and compactness.
 
|}
 
|}

Latest revision as of 15:57, 25 March 2023

Course name

MAT 3333 Fundamentals of Analysis and Topology.

Catalog entry: MAT 3333 Fundamentals of Analysis and Topology. Prerequisite: MAT 3003 Discrete Mathematics, or consent of instructor. Topological notions in the real line and in metric spaces. Convergent sequences. Continuous functions. Connected and compact sets. The Intermediate Value and Extreme Value theorems. Sequential compactness and the Heine-Borel Theorem.

Prerequisites: MAT 1224 and MAT 3003.

Sample textbooks:


Topics List

(Section numbers refer to Erdman's book.)

Week Sections Topics Student Learning Outcomes

1-3

Chapters 1 & 2. Appendices C, G & H.

Operations, order and intervals of the real line. Completeness of the real line. Suprema and infima. Basic topological notions in the real line.

  • Arithmetic operations of ℝ.
  • Field axioms.
  • Order of ℝ.
  • Intervals: open, closed, bounded and unbounded.
  • Upper and lower bounds of subsets of ℝ.
  • Least upper (supremum) and greatest lower (infimum) bound of a subset of ℝ.
  • The Least Upper Bound Axiom (completeness of ℝ).
  • The Archimedean property of ℝ.
  • Distance.
  • Neighborhoods and interior of a set.
  • Open subsets of ℝ.
  • Closed subsets of ℝ.

4

Chapter 3

Continuous functions on ℝ.

  • Continuity at a point (local continuity).
  • Continuous functions on ℝ (global continuity).
  • Continuous functions on subsets of ℝ.

4-5

Chapter 4

Convergence of real sequences. The Cauchy criterion. Subsequences.

  • Sequences in ℝ.
  • Convergent sequences.
  • Algebraic operations on convergent sequences.
  • Sufficient conditions for convergence. Cauchy criterion.
  • Subsequences.

6

Chapter 5

Connectedness and the Intermediate Value Theorem

  • Connected subsets of ℝ.
  • Continuous images of connected sets.
  • The Intermediate Value Theorem.

7

Chapter 6

Compactness and the Extreme Value Theorem.

  • Compact subsets of the real line.
  • Examples of compact subsets.
  • The Extreme Value Theorem.

8

Chapter 7

Limits of real functions.

  • Limit of a real function at a point.
  • Continuity and limits.
  • Arithmetic properties of limits.

9 & 10

Chapters 9, 10, 11

The topology of metric spaces.

  • Metric spaces. Examples.
  • Equivalent metrics.
  • Interior, closure, and boundary.
  • Accumulation point.
  • Boundary point.
  • Closure.
  • Open and closed sets.
  • The relative topology.

11

Chapter 12

Sequences in metric spaces.

  • Convergent sequences.
  • Sequential characterizations of topological properties.

12

Chapter 14

Continuity and limits.

  • Continuous functions between metric spaces.
  • Topological products.
  • Limits.

13

15.1–15.2

Compact metric spaces.

  • Compactness: definition and elementary properties.
  • The Extreme Value Theorem.

14

Chapter 16

Sequential compactness and the Heine-Borel Theorem.

  • Sequential compactness.
  • Conditions equivalent to compactness of a metric space.
  • The Heine-Borel Theorem.

15

Chapter 18 (time permitting)

Complete metric spaces.

  • Cauchy sequences in metric spaces.
  • Metric completness.
  • Completeness and compactness.
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