This course is an exercise session for the lecture course `Set and Topology II' (ZUA.B203). The materials for exercise are chosen from that course.
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|
Students are given exercise problems related to what is taught in the course "Set and Topology II"
|Course schedule||Required learning|
|Class 1||discussion session on the following materials: topology and topological space||Details will be provided during each class session|
|Class 2||discussion session on the following materials: open basis, system of neighborhoods, second countability|
|Class 3||discussion session on the following materials: fundamental system of neighborhoods, first countability|
|Class 4||discussion session on the following materials: continuous map, homeomorphism|
|Class 5||discussion session on the following materials: relative topology, product topology|
|Class 6||discussion session on the following materials: quotient topology, induced topology|
|Class 7||discussion session on the following materials: Hausdorff space, normal space|
|Class 8||evaluation of progress|
|Class 9||discussion session on the following materials: separation axioms and continuous functions|
|Class 10||discussion session on the following materials: connectedness of a topological space|
|Class 11||discussion session on the following materials: path-connectedness of a topological space|
|Class 12||discussion session on the following materials: compactness of a topological space|
|Class 13||discussion session on the following materials: properties of a compact space|
|Class 14||discussion session on the following materials: completeness of metric spaces|
|Class 15||discussion session on the following materials: topological properties of metric spaces|
Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
brief exam (about 30%), oral presentation for exercise problems (about 70%)
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.B203 ： Set and Topology II (if not passed yet) at the same time