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	Algebraic Structure of the Real Numbers	Order of Operations	Review of algebraic properties of the Real Numbers
Week 1	30-31	Distance Between Two Points in the Real Numbers	Absolute Value Algebraic Structure of the Real Numbers	The distance between two point on the real line
Week 1	30-31	Limit of a sequence in the Real Numbers	Distance Between Two Points in the Real Numbers Suprema, Infima, and the Completeness Property	The definition of a limit of a sequence
Week 1	30-31	The Nested Interval Theorem for the Real Numbers	Limit of a sequence in the Real Numbers Intervals Suprema, Infima, and the Completeness Property Mathematical Induction	The nested intervals property in the real numbers
Week 2	34-38	Cauchy-Schwarz Formula	Absolute Value Real Numbers:Inequalities	The Cauchy-Schwarz Formula
Week 2	34-38	Distance Between Two Points in Higher Dimensions	Distance Between Two Points in the Real Numbers	Distance formula for higher dimensions
Week 2	34-38	Open Subsets in Higher Dimensions	Sets:Operations Open Intervals Suprema, Infima, and the Completeness Property	Definition of an open set Definition of an interior point
Week 2	34-38	Limit Points (or Cluster Points) in Higher Dimensions	Open Subsets in Higher Dimensions Limit of a sequence in the Real Numbers Mathematical Induction	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	Sets:Operations Limit Points (or Cluster Points) in Higher Dimensions Open Subsets in Higher Dimensions	Definition of a closed set Properties of a closed set
Week 3	39-40	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 3	39-40	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 3	39-40	The Bolzano-Weierstrass Theorem in Higher Dimensions	The Bolzano-Weierstrass Theorem Bounded sets in Higher Dimensions	The Bolzano Weierstrass Theorem in higher dimensions
Week 4	41-43	Convergent Sequences and the Cauchy Criterion in Higher Dimension	Open Subsets in Higher Dimensions Limit of a sequence in the Real Numbers The Cauchy Criterion for Convergence	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	Open Subsets in Higher Dimensions Closed and Bounded sets	Definition of an open cover for a set Definition of compactness The Heine-Borel Theorem
Week 4	41-43	Lindeloff Theorem	Heine-Borel Theorem Intervals Unions and Intersections of Sets	The Lindeloff Theorem
Week 5	45 and 75-76	Topological Spaces	Unions and Intersections of Sets 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 5	45 and 75-76	Distance Functions, Metrics	Functions Triangle Inequality Bounded sets in Higher Dimensions	Criteria for a function to be a distance function Common distance functions
Week 5	45 and 75-76	Metric Spaces	Distance Functions, Metrics	The definition of a metric space Basic examples of metric spaces
Week 6	46-47	Open Sets and Closed Sets in Metric Spaces	Sets:Operations 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 7	48-49	A Topology Given By A Metric	Distance Functions, Metrics Open Sets and Closed Sets in Metric Spaces	Typologies given by metrics
Week 7	48-49	Subspaces of Metric Spaces	Open Sets and Closed Sets in Metric Spaces Metric Spaces	Subspace topologies in metric spaces
Week 8	50-52	Convergent Sequences in Metric Spaces	Sequences Metric Spaces	The Definition and basic properties of convergent sequences in a metric space
Week 8	50-52	Cartesian Products of Metric Spaces	The Cartiesian Product Open Sets and Closed Sets in Metric Spaces	Product topologies
Week 8 and 9	50-52	Continuous Mappings Between Metric Spaces	Open Subsets in Higher Dimensions Closed Subsets in Higher Dimensions Continuous Functions	Definition and basic properties of continuous mapping between metric spaces
Week 9	53-55	Separation Properties	Topological Spaces Open Sets and Closed Sets in Metric Spaces	Definition and basic examples of separations of a set
Week 9 and 10	53-55	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 10	56-59	Separable Metric Spaces	Separation Properties Metric Spaces Convergent Sequences in Metric Spaces	Definition and basic properties of continuous mapping between metric spaces
Week 10	56-59	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 10	56-59	Bounded Sets and Bounded Functions in a Metric Space	Totally Bounded Metric Spaces Functions:Range	Basic properties of bounded sets and bounded functions in metric spaces
Week 11	60-62	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 12		Review
Week 13	63-65	Complete Metric Spaces	The Cauchy Criterion Convergent Sequences in Metric Spaces Bounded Sets and Bounded Functions in a Metric Space	Definition and basic properties of complete metric spaces
Week 13	63-65	Baire's Theorem and Applications	Open Sets and Closed Sets in Metric Spaces Separation Properties The Derivative The Riemann Integral Monotone Functions	Baire's Theorem for metric spaces Applications of Baire's Theorem for metric spaces
Week 14	Edit	Stone-Weierstrass Theorem	Taylor and Maclaurin Series Compactness in Metric Spaces Continuous Mappings Between Metric Spaces	The Stone-Weierstrass Theorem
Week 15	70-71	The Hilbert Space L2 and the Hilbert Cube	Compactness in Metric Spaces Open Sets and Closed Sets in Metric Spaces Infinite Series	Basic Topological Properties of l2.