The main subject of this course is basic concepts in general topology. First, general topology on sets will be introduced in terms of system of open sets, closed sets, and system of neighborhoods, and then continuity for mapping between topological spaces will be discussed in terms of these notions. Next we explain various natural topology, such as metric topology, relative topology, quotient topology and product topology. Then we discuss various axioms of separability, such as Hausdorff property. Main subjects in the latter half of this course 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. We strongly recommend to take this course with "Exercises in Geometry A".
The notion of topological space is essential for describing continuity of mappings. It is significant not only in geometry but also algebra and analysis. Compactness and connectedness are most significant geometric properties of the space. They will be fundamental when learning more advanced geometry, such as theory of manifolds. Completeness and boundedness of metric spaces are fundamental concepts especially in analysis.
Students are expected to
・Understand various equivalent definitions of topology
・Understand that continuity of maps between topological spaces is described in terms of topology
・Understand various kinds of topologies that naturally arise under various settings
・Understand various separation axioms, with various examples
・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
topology and topological space, neighborhood, first countability, second countability, continuous mapping, induced topology, separation axioms, 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
Course schedule | Required learning | |
---|---|---|
Class 1 | topology and topological space | Details will be provided during each class session |
Class 2 | open basis, system of neighborhoods, second countability | Details will be provided during each class session |
Class 3 | fundamental system of neighborhoods, first countability | Details will be provided during each class session |
Class 4 | continuous map, homeomorphism | Details will be provided during each class session |
Class 5 | relative topology, product topology | Details will be provided during each class session |
Class 6 | quotient topology, induced topology | Details will be provided during each class session |
Class 7 | Hausdorff space, normal space | Details will be provided during each class session |
Class 8 | evaluation of progress | Details will be provided during each class session |
Class 9 | separation axioms and continuous functions | Details will be provided during each class session |
Class 10 | connectedness of a topological space | Details will be provided during each class session |
Class 11 | path-connectedness of a topological space | Details will be provided during each class session |
Class 12 | compactness of a topological space | Details will be provided during each class session |
Class 13 | properties of a compact space | Details will be provided during each class session |
Class 14 | completeness of metric spaces | Details will be provided during each class session |
Class 15 | topological properties of metric spaces | Details will be provided during each class session |
None required
Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
midterm exam (about 50%), final exam (about 50%)
Students are expected to have passed [Calculus I / Recitation], Calculus II + Recitation, [Linear Algebra I / Recitation] and Linear Algebra II + Recitation.
Strongly recommended to take ZUA.B204 ： Exercises in Geometry A (if not passed yet) at the same time