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:'''
Line 23: Line 23:
 
|
 
|
 
<!-- 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 -->
Line 35: Line 37:
 
* Field axioms.
 
* Field axioms.
 
* Order of ℝ.
 
* 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.
 +
 +
|-
 +
<!-- Week # -->
 +
|
 +
4
 +
||
 +
<!-- Sections -->
 +
Chapter 3
 +
||
 +
<!-- Topics -->
 +
Continuous functions on ℝ.
 +
||
 +
<!-- SLOs -->
 +
* Continuity at a point (local continuity).
 +
* Continuous functions on ℝ (global continuity).
 +
* Continuous functions on subsets of ℝ.
  
 
|-
 
|-
 
|
 
|
 
<!-- Week # -->
 
<!-- Week # -->
−
2
+
4-5
 
||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
1.2. Appendix J.
+
Chapter 4
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
−
Completeness of the real line. Suprema and infima.
+
Convergence of real sequences.
 +
The Cauchy criterion. Subsequences.
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
−
* Intervals: open, closed, bounded and unbounded.
+
* Sequences in ℝ.
−
* Upper and lower bounds of subsets of ℝ.
+
* Convergent sequences.
−
* Least upper (supremum) and greatest lower (infimum) bound of a subset of ℝ.
+
* Algebraic operations on convergent sequences.
−
* The Least Upper Bound Axiom (completeness of ℝ).
+
* Sufficient conditions for convergence. Cauchy criterion.
−
* The Archimedean property of ℝ.
+
* Subsequences.
  
 
|-
 
|-
 
|
 
|
−
3
+
<!-- Week # -->
 +
6
 
||
 
||
−
2.1, 2.2
+
<!-- Sections -->
 +
Chapter 5
 
||
 
||
−
Basic topological notions in the real line.
+
<!-- Topics -->
 +
Connectedness and the Intermediate Value Theorem
 
||
 
||
−
* Distance.
+
<!-- SLOs -->
−
* Neighborhoods and interior of a set.
+
* Connected subsets of ℝ.
−
* Open subsets of ℝ.
+
* Continuous images of connected sets.
−
* Closed subsets of ℝ.
+
* The Intermediate Value Theorem.
  
 
|-
 
|-
 +
|
 
<!-- Week # -->
 
<!-- Week # -->
 +
7
 +
||
 +
<!-- Sections -->
 +
Chapter 6
 +
||
 +
<!-- Topics -->
 +
Compactness and the Extreme Value Theorem.
 +
||
 +
<!-- SLOs -->
 +
* Compact subsets of the real line.
 +
* Examples of compact subsets.
 +
* The Extreme Value Theorem.
 +
 +
|-
 
|
 
|
−
4
+
<!-- Week # -->
 +
8
 
||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
3.1–3.3
+
Chapter 7
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
−
Continuous functions on subsets of the real line.
+
Limits of real functions.
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
−
* Continuity at a point (local continuity).
+
* Limit of a real function at a point.
−
* Continuous functions on ℝ (global continuity).
+
* Continuity and limits.
−
* Continuous functions on subsets of ℝ.
+
* Arithmetic properties of limits.
  
 
|-
 
|-
 +
|
 
<!-- Week # -->
 
<!-- Week # -->
−
5
+
9 & 10
 
||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
---
+
Chapters 9, 10, 11
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
−
Review. First midterm exam.
+
The topology of metric spaces.
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
 +
* Metric spaces. Examples.
 +
* Equivalent metrics.
 +
* Interior, closure, and boundary.
 +
* Accumulation point.
 +
* Boundary point.
 +
* Closure.
 +
* Open and closed sets.
 +
* The relative topology.
  
 
|-
 
|-
 +
|
 
<!-- Week # -->
 
<!-- Week # -->
−
6
+
11
 
||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
4.1, 4.2
+
Chapter 12
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
−
Convergence of real sequences.
+
Sequences in metric spaces.
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
−
* Sequences in ℝ.
 
 
* Convergent sequences.
 
* Convergent sequences.
−
* Algebraic operations on convergent sequences.
+
* Sequential characterizations of topological properties.
 +
 
 +
|-
 +
|
 +
<!-- Week # -->
 +
12
 +
||
 +
<!-- Sections -->
 +
Chapter 14
 +
||
 +
<!-- Topics -->
 +
Continuity and limits.
 +
||
 +
<!-- SLOs -->
 +
* Continuous functions between metric spaces.
 +
* Topological products.
 +
* Limits.
  
 
|-
 
|-
 +
|
 
<!-- Week # -->
 
<!-- Week # -->
−
7
+
13
 
||
 
||
 
<!-- Sections -->
 
<!-- Sections -->
−
4.3, 4.4
+
15.1–15.2
 
||
 
||
 
<!-- Topics -->
 
<!-- Topics -->
−
The Cauchy criterion. Subsequences.
+
Compact metric spaces.
 
||
 
||
 
<!-- SLOs -->
 
<!-- SLOs -->
−
* Sufficient conditions for convergence. Cauchy criterion.
+
* Compactness: definition and elementary properties.
−
* Subsequences.
+
* The Extreme Value Theorem.
  
 +
|-
 +
|
 +
<!-- Week # -->
 +
14
 +
||
 +
<!-- Sections -->
 +
Chapter 16
 +
||
 +
<!-- Topics -->
 +
Sequential compactness and the Heine-Borel Theorem.
 +
||
 +
<!-- SLOs -->
 +
* Sequential compactness.
 +
* Conditions equivalent to compactness of a metric space.
 +
* 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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