### 2018　Exercises in Set and Topology

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Mathematics
Instructor(s)
Endo Hisaaki  Kawai Shingo
Class Format
Exercise
Media-enhanced courses
Day/Period(Room No.)
Tue5-8(H103)  Tue7-10(H103)
Group
-
Course number
ZUA.B202
Credits
2
2018
Offered quarter
1-2Q
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 I' (ZUA.B201). The materials for exercise are chosen from that course.

### Student learning outcomes

Students are expected to
・Understand De Morgan’s law
・Be familiar with injectivity, surjectivity, and bijectivity of mappings
・Be able to determine the image and preimage of concrete maps
・Be familiar with many basic examples of equivalence relations and quotient sets
・Understand the difference between countable and uncountable sets
・Distinguish between semi-ordered sets and totally-ordered sets
・Be able to deduce strong properties of well-ordered sets
・Understand a few applications of Zorn’s lemma
・Understand equivalence between the well-ordering theorem, Zorn’s lemma and the axiom of choice
・Understand basic properties of Euclidean space and general metric spaces

### Keywords

set, map, image and inverse image, product set, binary relation, equivalence relation, quotient set, cardinality of sets, countable and uncountable set, ordered set, totally ordered set, well-ordered set, Zorn’s lemma, the axiom of choice, well-ordering theorem, Euclidean space, metric space, continuous map

### 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 I"

### Course schedule/Required learning

Course schedule Required learning
Class 1 discussion session on the following materials: examples of sets, union, intersection and subset, complement Details will be provided during each class session
Class 2 discussion session on the following materials: De Morgan's law, distributive law, mapping between sets Details will be provided during each class session
Class 3 discussion session on the following materials: image and preimage of map, composition of maps, product set Details will be provided during each class session
Class 4 discussion session on the following materials: correspondence between sets, indexed set Details will be provided during each class session
Class 5 discussion session on the following materials: binary relation, equivalence relation, equivalence class, quotient set Details will be provided during each class session
Class 6 discussion session on the following materials: the cardinality of set, relation between cardinality, countable set Details will be provided during each class session
Class 7 discussion session on the following materials: cardinality of the continuum, uncountable set, cardinality of power set 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: order, total order, well-ordered set and their basic properties Details will be provided during each class session
Class 10 discussion session on the following materials: inductive set, Zorn's lemma Details will be provided during each class session
Class 11 discussion session on the following materials: ordinal number, comparison of cardinality Details will be provided during each class session
Class 12 discussion session on the following materials: Equivalence between the well-ordering theorem, Zorn’s lemma and the axiom of choice Details will be provided during each class session
Class 13 discussion session on the following materials: Application of Zorn's lemma Details will be provided during each class session
Class 14 discussion session on the following materials: Euclidean space, metric space, open set and closed set Details will be provided during each class session
Class 15 discussion session on the following materials: basic concepts on 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.