### 2018　Exercises in Geometry A

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Academic unit or major
Mathematics
Instructor(s)
Endo Hisaaki  Kawai Shingo
Class Format
Exercise
Media-enhanced courses
Day/Period(Room No.)
Tue5-8(H103)
Group
-
Course number
ZUA.B204
Credits
2
Academic year
2018
Offered quarter
3-4Q
Syllabus updated
2018/3/20
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

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.

### Student learning outcomes

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

### Keywords

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

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

Students are given exercise problems related to what is taught in the course "Set and Topology II"

### Course schedule/Required learning

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 Details will be provided during each class session
Class 3 discussion session on the following materials: fundamental system of neighborhoods, first countability Details will be provided during each class session
Class 4 discussion session on the following materials: continuous map, homeomorphism Details will be provided during each class session
Class 5 discussion session on the following materials: relative topology, product topology Details will be provided during each class session
Class 6 discussion session on the following materials: quotient topology, induced topology Details will be provided during each class session
Class 7 discussion session on the following materials: 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 discussion session on the following materials: separation axioms and continuous functions Details will be provided during each class session
Class 10 discussion session on the following materials: connectedness of a topological space Details will be provided during each class session
Class 11 discussion session on the following materials: path-connectedness of a topological space Details will be provided during each class session
Class 12 discussion session on the following materials: compactness of a topological space Details will be provided during each class session
Class 13 discussion session on the following materials: properties of a compact space Details will be provided during each class session
Class 14 discussion session on the following materials: completeness of metric spaces Details will be provided during each class session
Class 15 discussion session on the following materials: topological properties of metric spaces Details will be provided during each class session

None required

### Reference books, course materials, etc.

Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.

### Assessment criteria and methods

brief exam (about 30%), oral presentation for exercise problems (about 70%)

### Related courses

• ZUA.B203 ： Set and Topology II
• MTH.B203 ： Introduction to Topology III
• MTH.B204 ： Introduction to Topology IV

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

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