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
|
|
|
|
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
|
|
|
|
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 8
|
50-52
|
Continuous Mappings Between Metric Spaces
|
|
- Definition and basic properties of continuous mapping between metric spaces
|
|
Week 8
|
50-52
|
Continuous Mappings Between Metric Spaces
|
|
- Definition and basic properties of continuous mapping between metric spaces
|
|