Difference between revisions of "MAT4273"

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(Added content to the table(weeks 1 and 2))
 
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* [[Order of Operations]] <!-- 1073-Mod R -->
 
* [[Order of Operations]] <!-- 1073-Mod R -->
−
 
  
 
||
 
||
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−
[[Distance Between Two Points in the Real Numbers]]  
+
[[Distance Between Two Points|Distance Between Two Points in the Real Numbers]]  
  
 
||
 
||
Line 72: Line 71:
 
||
 
||
  
−
* [[Distance Between Two Points in the Real Numbers]] <!-- 4237-(30-31) -->
+
* [[Distance Between Two Points|Distance Between Two Points in the Real Numbers]] <!-- 4237-(30-31) -->
 
* [[Suprema, Infima, and the Completeness Property]] <!-- 3213-2.3 -->  
 
* [[Suprema, Infima, and the Completeness Property]] <!-- 3213-2.3 -->  
  
Line 98: Line 97:
 
* [[Intervals ]] <!-- 3213-2.5 -->  
 
* [[Intervals ]] <!-- 3213-2.5 -->  
 
* [[Suprema, Infima, and the Completeness Property]] <!-- 3213-2.3 -->  
 
* [[Suprema, Infima, and the Completeness Property]] <!-- 3213-2.3 -->  
 +
* [[Proofs:Induction|Mathematical Induction]] <!-- 3213-1.2 -->
  
 
||
 
||
Line 118: Line 118:
 
    
 
    
  
−
[[Cauchy-Schwarz formula]]  
+
[[Cauchy-Schwarz Formula]]  
  
 
||
 
||
  
−
*  
+
* [[Absolute Value Functions| Absolute Value]] <!-- DNE (recommend 1073-1) -->
 +
* [[Real Numbers:Inequalities ]] <!-- 3213-2.1 -->
  
 
||
 
||
  
−
*  
+
* The Cauchy-Schwarz Formula
  
 
||
 
||
−
 
  
  
Line 143: Line 143:
 
||
 
||
 
    
 
    
−
 
+
[[Distance Between Two Points|Distance Between Two Points in Higher Dimensions]]  
−
[[Distance between Two Points in Higher Dimensions]]  
 
  
 
||
 
||
  
−
*  
+
* [[Distance Between Two Points|Distance Between Two Points in the Real Numbers]] <!-- 4237-(30-31) -->
  
 
||
 
||
  
−
*  
+
* Distance formula for higher dimensions
  
 
||
 
||
Line 168: Line 167:
 
||
 
||
 
    
 
    
−
 
 
[[Open Subsets in Higher Dimensions]]
 
[[Open Subsets in Higher Dimensions]]
  
 
||
 
||
  
−
*  
+
* [[Sets:Operations]] <!-- 3013- (3.1-3.3) -->
 +
* [[Intervals|Open Intervals ]] <!-- 3213-2.5 -->
 +
* [[Suprema, Infima, and the Completeness Property]] <!-- 3213-2.3 -->
  
 
||
 
||
  
−
*  
+
* Definition of an open set
 +
* Definition of an interior point
  
 
||  
 
||  
Line 190: Line 191:
  
 
<div style="text-align: center;">34-38</div>
 
<div style="text-align: center;">34-38</div>
 +
 +
|| 
 +
 +
[[Limit Points (or Cluster Points) in Higher Dimensions]]
 +
 +
||
 +
 +
* [[Open Subsets in Higher Dimensions]] <!-- 4237-(34-38) -->
 +
* [[Limit of a sequence in the Real Numbers]] <!-- 4237-(34-38) -->
 +
* [[Proofs:Induction|Mathematical Induction]] <!-- 3213-1.2 -->
 +
 +
||
 +
 +
* Definition of a limit point (or cluster point)
 +
* The limit point as the limit of a sequence
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;3
 +
 +
||
 +
 +
<div style="text-align: center;">39-40</div>
 +
 +
||
 +
 +
[[Closed Subsets in Higher Dimensions]]
 +
 +
||
 +
 +
* [[Sets:Operations]] <!-- 3013- (3.1-3.3) -->
 +
* [[Limit Points (or Cluster Points) in Higher Dimensions]]
 +
* [[Open Subsets in Higher Dimensions]]
 +
 +
||
 +
 +
* Definition of a closed set
 +
* Properties of a closed set
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;3
 +
 +
||
 +
 +
<div style="text-align: center;">39-40</div>
 +
 +
||
 +
 
 +
[[Bounded sets in Higher Dimensions]]
 +
 +
||
 +
 +
* [[Closed Subsets in Higher Dimensions]]
 +
* [[Limit Points (or Cluster Points) in Higher Dimensions]]
 +
* [[Open Subsets in Higher Dimensions]]
 +
 +
||
 +
 +
* Definition of a bounded set
 +
* Properties of a bounded set
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;3
 +
 +
||
 +
 +
<div style="text-align: center;">39-40</div>
 +
 +
||
 +
 
 +
[[The Nested Interval Theorem in Higher Dimensions]]
 +
 +
||
 +
 +
* [[The Nested Interval Theorem for the Real Numbers]]
 +
* [[Bounded sets in Higher Dimensions]]
 +
 +
||
 +
 +
* The nested intervals theorem in higher dimensions
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;3
 +
 +
||
 +
 +
<div style="text-align: center;">39-40</div>
 +
 +
||
 +
 
 +
[[Completeness|The Bolzano-Weierstrass Theorem in Higher Dimensions]]
 +
 +
||
 +
 +
* [[Theorem:Bolzano-Weierstrass|The Bolzano-Weierstrass Theorem]] <!-- 3213-3.4 -->
 +
* [[Bounded sets in Higher Dimensions]]
 +
 +
||
 +
 +
* The Bolzano Weierstrass Theorem in higher dimensions
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;4
 +
 +
||
 +
 +
<div style="text-align: center;">41-43</div>
 +
 +
|| 
 +
 +
[[Completeness|Convergent Sequences and the Cauchy Criterion in Higher Dimension]]
 +
 +
||
 +
 +
* [[Open Subsets in Higher Dimensions]]
 +
* [[Limit of a sequence in the Real Numbers]]
 +
* [[Series|The Cauchy Criterion for Convergence]] <!-- 3213-3.5 -->
 +
 +
||
 +
 +
* Definition of a convergent sequence in higher dimensions
 +
* The Cauchy criterion for convergence of sequences in higher dimensions
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;4
 +
 +
||
 +
 +
<div style="text-align: center;">41-43</div>
 +
 +
||
 +
 
 +
 +
[[Heine-Borel Theorem]]
 +
 +
||
 +
 +
* [[Open Subsets in Higher Dimensions]]
 +
* [[Intervals| Closed and Bounded sets]] <!-- 3213-2.5 -->
 +
 +
||
 +
 +
* Definition of an open cover for a set
 +
* Definition of compactness
 +
* The Heine-Borel Theorem
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;4
 +
 +
||
 +
 +
<div style="text-align: center;">41-43</div>
  
 
||
 
||
 
    
 
    
  
−
[[Limit Points (or Cluster Points) in Higher Dimensions]]
+
[[Lindeloff Theorem]]
 +
 
 +
||
 +
 
 +
* [[Heine-Borel Theorem]]
 +
* [[Intervals]] <!-- 3213-2.5 -->
 +
* [[Unions and Intersections of Sets]] <!-- DNE(recommend 3013) -->
 +
 
 +
||
 +
 
 +
* The Lindeloff Theorem
 +
 
 +
||
 +
 
 +
 
 +
|-
 +
 
 +
 
 +
|Week&nbsp;5
 +
 
 +
||
 +
 
 +
<div style="text-align: center;">45 and 75-76</div>
 +
 
 +
||
 +
 
 +
[[Topological Spaces]]
 +
 
 +
||
 +
 
 +
* [[Unions and Intersections of Sets]] <!-- DNE(recommend 3013) -->
 +
* [[Open Subsets in Higher Dimensions]]
 +
* [[Closed Subsets in Higher Dimensions]]
 +
 
 +
||
 +
 
 +
* Definition of a Topological space
 +
* Examples of Topological spaces
 +
* Basic theorems concerning topological spaces
 +
 
 +
||
 +
 
 +
 
 +
|-
 +
 
 +
 
 +
 
 +
|Week&nbsp;5
 +
 
 +
||
 +
 
 +
<div style="text-align: center;">45 and 75-76</div>
 +
 
 +
||
 +
 
 +
 
 +
[[Distance Functions, Metrics]]
  
 
||
 
||
  
−
*  
+
* [[Functions]] <!-- 1073-Mod 2.1 -->
 +
* [[Triangle Inequality]] <!-- DNE(recommend 3013) -->
 +
* [[Bounded sets in Higher Dimensions]]
  
 
||
 
||
  
−
*  
+
* Criteria for a function to be a distance function
 +
* Common distance functions
  
 
||
 
||
Line 208: Line 454:
  
 
|-
 
|-
 +
 +
  
  
 
|Week&nbsp;5
 
|Week&nbsp;5
 +
 +
||
 +
 +
<div style="text-align: center;">45 and 75-76</div>
 +
 +
||
 +
 
 +
[[Metric Spaces]]
 +
 +
||
 +
 +
* [[Distance Functions, Metrics]]
 +
 +
||
 +
 +
* The definition of a metric space
 +
* Basic examples of metric spaces
 +
 +
||
 +
 +
 +
 +
|-
 +
 +
 +
 +
|Week&nbsp;6
 +
 +
||
 +
 +
<div style="text-align: center;">46-47</div>
 +
 +
||
 +
 
 +
[[Open Sets and Closed Sets in Metric Spaces]]
 +
 +
||
 +
 +
* [[Sets:Operations]] <!-- 3013- (3.1-3.3) -->
 +
* [[Metric Spaces]]
 +
* [[Open Subsets in Higher Dimensions]]
 +
* [[Closed Subsets in Higher Dimensions]]
 +
 +
||
 +
 +
* Definition of an open set in a metric space
 +
* Definition of a closed set in a metric space
 +
* Basic examples of open and closed sets in metric spaces
 +
 +
||
 +
 +
 +
 +
|-
 +
 +
 +
 +
|Week&nbsp;7
 +
 +
||
 +
 +
<div style="text-align: center;">48-49</div>
 +
 +
||
 +
 
 +
[[A Topology Given By A Metric]]
 +
 +
||
 +
 +
* [[Distance Functions, Metrics]]
 +
* [[Open Sets and Closed Sets in Metric Spaces]]
 +
 +
||
 +
 +
* Typologies given by metrics
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;7
 +
 +
||
 +
 +
<div style="text-align: center;">48-49</div>
 +
 +
||
 +
 
 +
[[Subspaces of Metric Spaces]]
 +
 +
||
 +
 +
* [[Open Sets and Closed Sets in Metric Spaces]]
 +
* [[Metric Spaces]]
 +
 +
||
 +
 +
* Subspace topologies in metric spaces
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;8
 +
 +
||
 +
 +
<div style="text-align: center;">50-52</div>
 +
 +
||
 +
 
 +
 +
[[Convergent Sequences in Metric Spaces]]
 +
 +
||
 +
 +
* [[Limit of a Sequence in the Real Numbers|Sequences]] <!-- 4213-3.1,  also listed earlier in this course-->
 +
* [[Metric Spaces]]
 +
 +
||
 +
 +
* The Definition and basic properties of convergent sequences in a metric space
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;8
 +
 +
||
 +
 +
<div style="text-align: center;">50-52</div>
 +
 +
||
 +
 
 +
[[Cartesian Products of Metric Spaces]]
 +
 +
||
 +
 +
* [[The Cartiesian Product]] <!-- 4213-1.1-->
 +
* [[Bases of Open Sets|Open Sets and Closed Sets in Metric Spaces]]
 +
 +
||
 +
 +
* Product topologies
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;8 and 9
 +
 +
||
 +
 +
<div style="text-align: center;">50-52</div>
 +
 +
|| 
 +
 +
[[Continuous Mappings Between Metric Spaces]]
 +
 +
||
 +
 +
* [[Open Subsets in Higher Dimensions]]
 +
* [[Closed Subsets in Higher Dimensions]]
 +
* [[Continuous Functions]] <!-- 4213 - 5.1 -->
 +
 +
||
 +
 +
* Definition and basic properties of continuous mapping between metric spaces
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;9
 +
 +
||
 +
 +
<div style="text-align: center;">53-55</div>
 +
 +
|| 
 +
 +
[[Separation Properties]]
 +
 +
||
 +
 +
* [[Topological Spaces]]
 +
* [[Open Sets and Closed Sets in Metric Spaces]]
 +
 +
||
 +
 +
* Definition and basic examples of separations of a set
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;9 and 10
 +
 +
||
 +
 +
<div style="text-align: center;">53-55</div>
 +
 +
|| 
 +
 +
[[Connectedness]]
 +
 +
||
 +
 +
* [[Open Sets and Closed Sets in Metric Spaces]]
 +
* [[Continuous Mappings Between Metric Spaces]]
 +
 +
||
 +
 +
* Definition and basic properties of connected sets
 +
* Invariance of connectedness under continuous mappings
 +
* Path/polygonal connectedness
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;10
 +
 +
||
 +
 +
<div style="text-align: center;">56-59</div>
 +
 +
|| 
 +
 +
[[Separable Metric Spaces]]
 +
 +
||
 +
 +
* [[Separation Properties]]
 +
* [[Metric Spaces]]
 +
* [[Convergent Sequences in Metric Spaces]]
 +
 +
||
 +
 +
* Definition and basic properties of continuous mapping between metric spaces
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;10
 +
 +
||
 +
 +
<div style="text-align: center;">56-59</div>
 +
 +
|| 
 +
 +
[[Totally Bounded Metric Spaces]]
 +
 +
||
 +
 +
* [[Metric Spaces]]
 +
* [[Open Sets and Closed Sets in Metric Spaces]]
 +
* [[Bounded sets in Higher Dimensions]]
 +
 +
||
 +
 +
* Definition and basic properties of separable and totally bounded metric spaces
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;10
 +
 +
||
 +
 +
<div style="text-align: center;">56-59</div>
 +
 +
|| 
 +
 +
[[Bounded Sets and Bounded Functions in a Metric Space]]
 +
 +
||
 +
 +
* [[Totally Bounded Metric Spaces]]
 +
* [[Functions:Range]] <!-- 4213 - 1.1 -->
 +
 +
||
 +
 +
* Basic properties of bounded sets and bounded functions in metric spaces
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;11
 +
 +
||
 +
 +
<div style="text-align: center;">60-62</div>
 +
 +
|| 
 +
 +
[[Compactness in Metric Spaces]]
 +
 +
||
 +
 +
* [[Continuous Mappings Between Metric Spaces]]
 +
* [[Open Sets and Closed Sets in Metric Spaces]]
 +
* [[Bounded Sets and Bounded Functions in a Metric Space]]
 +
 +
||
 +
 +
* Definition and basic properties of compact sets in metric spaces
 +
* Invariance of compactness under continuous mappings
 +
 +
||
 +
 +
 +
|-
 +
 +
 +
|Week&nbsp;12
  
 
||
 
||
  
 
<div style="text-align: center;"></div>
 
<div style="text-align: center;"></div>
 +
 +
|| 
 +
 +
'''Review'''
 +
 +
||
 +
 +
  
 
||
 
||
−
 
 
  
−
[[]]
+
 
 +
 
 +
||
 +
 
 +
 
 +
|-
 +
 
 +
 
 +
|Week&nbsp;13
 +
 
 +
||
 +
 
 +
<div style="text-align: center;">63-65</div>
 +
 
 +
|| 
 +
 
 +
[[Complete Metric Spaces]]
 +
 
 +
||
 +
 
 +
* [[The Cauchy Criterion]] <!-- 3213 - 3.5 -->
 +
* [[Convergent Sequences in Metric Spaces]]
 +
* [[Bounded Sets and Bounded Functions in a Metric Space]]
 +
 
 +
||
 +
 
 +
* Definition and basic properties of complete metric spaces
 +
 
 +
||
 +
 
 +
 
 +
|-
 +
 
 +
 
 +
|Week&nbsp;13
 +
 
 +
||
 +
 
 +
<div style="text-align: center;">63-65</div>
 +
 
 +
|| 
 +
 
 +
[[Baire's Theorem and Applications]]
 +
 
 +
||
 +
 
 +
* [[Open Sets and Closed Sets in Metric Spaces]]
 +
* [[Separation Properties]]
 +
* [[The Derivative]]  <!-- 4213 - 6.1 --> 
 +
* [[The Riemann Integral]]  <!-- 4213 - 7.1 --> 
 +
* [[Monotone Functions]] <!-- 4213 - 5.6 -->
 +
 
 +
||
 +
 
 +
* Baire's Theorem for metric spaces
 +
* Applications of Baire's Theorem for metric spaces
 +
 
 +
||
 +
 
 +
 
 +
|-
 +
 
 +
 
 +
|Week&nbsp;14
 +
 
 +
||
 +
 
 +
<div style="text-align: center;">Edit</div>
 +
 
 +
|| 
 +
 
 +
[[Stone-Weierstrass Theorem]]
 +
 
 +
||
 +
 
 +
* [[Taylor and Maclaurin Series]] <!-- 1224 - 6.3 -->
 +
* [[Compactness in Metric Spaces]]
 +
* [[Continuous Mappings Between Metric Spaces]]
 +
 
 +
||
 +
 
 +
* The Stone-Weierstrass Theorem
 +
 
 +
||
 +
 
 +
 
 +
|-
 +
 
 +
 
 +
|Week&nbsp;15
 +
 
 +
||
 +
 
 +
<div style="text-align: center;">70-71</div>
 +
 
 +
|| 
 +
 
 +
[[The Hilbert Space L2 and the Hilbert Cube]]
  
 
||
 
||
  
−
*  
+
* [[Compactness in Metric Spaces]]
 +
* [[Open Sets and Closed Sets in Metric Spaces]]
 +
* [[Infinite Series]] <!-- 1224 - 5.2 -->
 +
 
 
||
 
||
  
−
*  
+
* Basic Topological Properties of l<sup>2</sup>.
  
 
||
 
||

Latest revision as of 15:09, 23 January 2022

The textbook(s) for this course is [1] Undergraduate Topology, by R. H. Kasriel; topics from [2] Introduction to Analysis by W. R. Wade (Stone-Weierstrass Theorem).

A comprehensive list of all undergraduate math courses at UTSA can be found here.

The Wikipedia summary of topology and its history.

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1
30-31

Algebraic Structure of the Real Numbers

  • Review of algebraic properties of the Real Numbers


Week 1
30-31


Distance Between Two Points in the Real Numbers

  • The distance between two point on the real line
Week 1
30-31

Limit of a sequence in the Real Numbers

  • The definition of a limit of a sequence


Week 1
30-31

The Nested Interval Theorem for the Real Numbers

  • The nested intervals property in the real numbers


Week 2
34-38


Cauchy-Schwarz Formula

  • The Cauchy-Schwarz Formula


Week 2
34-38

Distance Between Two Points in Higher Dimensions

  • Distance formula for higher dimensions


Week 2
34-38

Open Subsets in Higher Dimensions

  • Definition of an open set
  • Definition of an interior point


Week 2
34-38

Limit Points (or Cluster Points) in Higher Dimensions

  • Definition of a limit point (or cluster point)
  • The limit point as the limit of a sequence


Week 3
39-40

Closed Subsets in Higher Dimensions

  • Definition of a closed set
  • Properties of a closed set


Week 3
39-40

Bounded sets in Higher Dimensions

  • Definition of a bounded set
  • Properties of a bounded set


Week 3
39-40

The Nested Interval Theorem in Higher Dimensions

  • The nested intervals theorem in higher dimensions


Week 3
39-40

The Bolzano-Weierstrass Theorem in Higher Dimensions

  • The Bolzano Weierstrass Theorem in higher dimensions


Week 4
41-43

Convergent Sequences and the Cauchy Criterion in Higher Dimension

  • Definition of a convergent sequence in higher dimensions
  • The Cauchy criterion for convergence of sequences in higher dimensions


Week 4
41-43


Heine-Borel Theorem

  • Definition of an open cover for a set
  • Definition of compactness
  • The Heine-Borel Theorem


Week 4
41-43


Lindeloff Theorem

  • The Lindeloff Theorem


Week 5
45 and 75-76

Topological Spaces

  • Definition of a Topological space
  • Examples of Topological spaces
  • Basic theorems concerning topological spaces


Week 5
45 and 75-76


Distance Functions, Metrics

  • Criteria for a function to be a distance function
  • Common distance functions


Week 5
45 and 75-76

Metric Spaces

  • The definition of a metric space
  • Basic examples of metric spaces


Week 6
46-47

Open Sets and Closed Sets in Metric Spaces

  • Definition of an open set in a metric space
  • Definition of a closed set in a metric space
  • Basic examples of open and closed sets in metric spaces


Week 7
48-49

A Topology Given By A Metric

  • Typologies given by metrics


Week 7
48-49

Subspaces of Metric Spaces

  • Subspace topologies in metric spaces


Week 8
50-52


Convergent Sequences in Metric Spaces

  • The Definition and basic properties of convergent sequences in a metric space


Week 8
50-52

Cartesian Products of Metric Spaces

  • Product topologies


Week 8 and 9
50-52

Continuous Mappings Between Metric Spaces

  • Definition and basic properties of continuous mapping between metric spaces


Week 9
53-55

Separation Properties

  • Definition and basic examples of separations of a set


Week 9 and 10
53-55

Connectedness

  • Definition and basic properties of connected sets
  • Invariance of connectedness under continuous mappings
  • Path/polygonal connectedness


Week 10
56-59

Separable Metric Spaces

  • Definition and basic properties of continuous mapping between metric spaces


Week 10
56-59

Totally Bounded Metric Spaces

  • Definition and basic properties of separable and totally bounded metric spaces


Week 10
56-59

Bounded Sets and Bounded Functions in a Metric Space

  • Basic properties of bounded sets and bounded functions in metric spaces


Week 11
60-62

Compactness in Metric Spaces

  • Definition and basic properties of compact sets in metric spaces
  • Invariance of compactness under continuous mappings


Week 12

Review




Week 13
63-65

Complete Metric Spaces

  • Definition and basic properties of complete metric spaces


Week 13
63-65

Baire's Theorem and Applications

  • Baire's Theorem for metric spaces
  • Applications of Baire's Theorem for metric spaces


Week 14
Edit

Stone-Weierstrass Theorem

  • The Stone-Weierstrass Theorem


Week 15
70-71

The Hilbert Space L2 and the Hilbert Cube

  • Basic Topological Properties of l2.


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