Math 46300/A6300: Course Learning Outcomes
- Department: Mathematics
- Course number: Math 46300/A6300
- Course title: Topology
- Term offered: Spring 2025
- Prerequisites: Math 32404
- Hours and credits: 4 hours and 4 credits
- Date effective: January 25, 2025
Catalog description
A course in general topology. Topological spaces: metric spaces, subspaces, continuous maps, connectedness, separation axioms; topological vector spaces: Hilbert spaces, Banach space, Frechet spaces; the quotient topology or identification spaces: the classification of two-dimensional manifolds; fundamental group and covering spaces; covering spaces of graphs: applications to group theory.
Textbooks used
- Munkres. Topology, 2nd edition, Prentice Hall, 2000.
Other sources such as exerpts from other textbooks, or parts of other freely available textbooks may be used as well.
Topics covered
I tentatively plan to cover at least the following topics:
- Munkres §12: Topological Spaces
- Munkres §13: Basis for a Topology
- Munkres §16: The Subspace Topology
- Munkres §17: Closed Sets and Limit Points
- Munkres §18: Continuous Functions
- Munkres §19: The Product Topology
- Munkres §20: The Metric Topology
- Munkres §22: The Quotient Topology
- Munkres §9: Infinite Sets and the Axiom of Choice
- Munkres §23: Connected Spaces
- Munkres §26: Compact Spaces
- Munkres §28: Limit Point Compactness
- Munkres §29: Local Compactness
- Munkres §30: The Countability Axioms
- Munkres §31: The Separation Axioms
- Munkres §51: Homotopy of Paths
- Munkres §52: The Fundamental Group
- Munkres §53: Covering Spaces
- Munkres §54: The Fundamental Group of the Circle
- Munkres §60: Fundamental Groups of Some Surfaces
Course assessment tools
- Homework
- Classwork
- Quizzes
- Midterm exams
- The final exam
See the syllabus for more details including information on how your course grade will be computed.
Course learning outcomes:
| After taking this course, the students should be able to: | Contributes to Departmental learning outcome(s): |
|---|---|
| 1) Write clear and rigorous proofs (or disproofs) of mathematical statements concerning general topology. | e1, e2, f, g |
| 2) Understand basic definitions and properties of topological spaces, such as connectedness, compactness, the various separation properties, the fundamental group and covering spaces. | e1, f, g |
| 3) Compute the fundamental group of various simply presented spaces. | a, d, e1, e2 |
| 4) Understand and work with various examples of topological spaces, including spaces constructed by various means including by subspaces, products, and topological quotients. | a, b, c, e2, g |
Departmental learning outcomes:
The mathematics department, in its varied courses, aims to teach students to:
- a) perform numeric and symbolic computations
- b) construct and apply symbolic and graphical representations of functions
- c) model real-life problems mathematically
- d) use technology appropriately to analyze mathematical problems
- e) state (e1) and apply (e2) mathematical definitions and theorems
- f) prove fundamental theorems
- g) construct and present (generally in writing, but, occasionally, orally) a rigorous mathematical argument.