# MAT4273

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
• Review of algebraic properties of the Real Numbers

Week 1
30-31
• The distance between two point on the real line
Week 1
30-31
• The definition of a limit of a sequence

Week 1
30-31
• The nested intervals property in the real numbers

Week 2
34-38
• The Cauchy-Schwarz Formula

Week 2
34-38
• Distance formula for higher dimensions

Week 2
34-38
• Definition of an open set
• Definition of an interior point

Week 2
34-38
• Definition of a limit point (or cluster point)
• The limit point as the limit of a sequence

Week 3
39-40
• Definition of a closed set
• Properties of a closed set

Week 3
39-40
• Definition of a bounded set
• Properties of a bounded set

Week 3
39-40
• The nested intervals theorem in higher dimensions

Week 3
39-40
• The Bolzano Weierstrass Theorem in higher dimensions

Week 4
41-43
• Definition of a convergent sequence in higher dimensions
• The Cauchy criterion for convergence of sequences in higher dimensions

Week 4
41-43
• Definition of an open cover for a set
• Definition of compactness
• The Heine-Borel Theorem

Week 4
41-43
• The Lindeloff Theorem

Week 5
45 and 75-76
• Definition of a Topological space
• Examples of Topological spaces
• Basic theorems concerning topological spaces

Week 5
45 and 75-76
• Criteria for a function to be a distance function
• Common distance functions

Week 5
45 and 75-76
• The definition of a metric space
• Basic examples of metric spaces

Week 6
46-47
• 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
• Typologies given by metrics

Week 7
48-49
• Subspace topologies in metric spaces

Week 8
50-52
• The Definition and basic properties of convergent sequences in a metric space

Week 8
50-52
• Product topologies

Week 8 and 9
50-52
• Definition and basic properties of continuous mapping between metric spaces

Week 9
53-55
• Definition and basic examples of separations of a set

Week 9 and 10
53-55
• Definition and basic properties of connected sets
• Invariance of connectedness under continuous mappings
• Path/polygonal connectedness

Week 10
56-59
• Definition and basic properties of continuous mapping between metric spaces

Week 10
56-59
• Definition and basic properties of separable and totally bounded metric spaces

Week 10
56-59
• Basic properties of bounded sets and bounded functions in metric spaces

Week 11
60-62
• Definition and basic properties of compact sets in metric spaces
• Invariance of compactness under continuous mappings

Week 12

Review

Week 13
63-65
• Definition and basic properties of complete metric spaces

Week 13
63-65
• Baire's Theorem for metric spaces
• Applications of Baire's Theorem for metric spaces

Week 14
Edit
• The Stone-Weierstrass Theorem

Week 15
70-71
• Basic Topological Properties of l2.