This course is a succession of “Introduction to Topology III” in 3Q. Main subjects are geometric properties of topological spaces, such as compactness, (path-) connectedness. Compact spaces have distinguished property that any function has maximum and minimum, and one of the fundamental properties of a space. A number of significant examples of compact/ non-compact and connected/disconnected spaces are provided. Also completeness and boundedness of metric spaces are treated.
Compactness and connectedness are most significant geometric properties of the space. They will be fundamental when learning more advanced geometry, such as manifolds. Completeness and boundedness are fundamental concepts especially in analysis.
Students are expected to
・Be able to prove basic properties of connected and compact spaces
・Learn a lot of basic examples of compact/ non-compact and connected/disconnected spaces
・Understand basic properties of complete metric spaces and examples
compact space, connected spaces, path-connectedness, completeness of a metric space
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course accompanied by discussion sessions
Course schedule | Required learning | |
---|---|---|
Class 1 | separation axioms and continuous functions | Details will be provided during each class session |
Class 2 | discussion session | |
Class 3 | connectedness of a topological space | |
Class 4 | discussion session | |
Class 5 | path-connectedness of a topological space | |
Class 6 | discussion session | |
Class 7 | compactness of a topological space | |
Class 8 | discussion session | |
Class 9 | properties of a compact space | |
Class 10 | discussion session | |
Class 11 | completeness of metric spaces | |
Class 12 | discussion session | |
Class 13 | topological properties of metric spaces | |
Class 14 | discussion session | |
Class 15 | evaluation of progress |
none required
Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
final exam 70%, discussion session 30%.
Required to have passed Introduction to Topology III.
Expected to have passed Introduction to Topology I and II.
Expected to have passed [Calculus I / Recitation], Calculus II + Recitation, [Linear Algebra I / Recitation] and Linear Algebra II + Recitation