### 2022　Introduction to Topology I

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Instructor(s)
Kalman Tamas  Kawai Shingo
Class Format
Lecture / Exercise    (Blended)
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(H103)  Tue7-10(H103)
Group
-
Course number
MTH.B201
Credits
2
2022
Offered quarter
1Q
Syllabus updated
2022/3/16
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

The main subject of this course is basic concepts in set theory. After introducing some fundamental operations for sets such as intersection, union and complement, we explain basic notions for maps between sets, such as injection, surjection, and bijection. Next we introduce binary relations on sets, especially the concept of equivalence relation and the associated quotient set. Finally, we introduce the equivalence of sets, and learn the notion of cardinality. Each lecture will be accompanied by a problem solving class. This course will be succeeded by “Introduction to Topology II” in the second quarter.
The notions of set and map are fundamental not only in mathematics but also in science, and are applicable to describe a wide variety of objects. On the other hand, these abstract notions are not easy to comprehend without suitable training. To that end, rigorous proofs will be provided for most propositions, lemmas and theorems.

### 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 maps
・Be familiar with many basic examples of equivalence relations and quotient sets
・Understand the difference between countable and uncountable sets

### Keywords

set, map, image and inverse image, product set, binary relation, equivalence relation, quotient set, cardinality of sets, countable and uncountable set

### Competencies that will be developed

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

### Class flow

Standard lecture course accompanied by discussion sessions

### Course schedule/Required learning

Course schedule Required learning
Class 1 examples of sets, union, intersection and subset, complement Details will be provided during each class session
Class 2 discussion session Details will be provided during each class session
Class 3 De Morgan's law, distributive law, mapping between sets Details will be provided during each class session
Class 4 discussion session Details will be provided during each class session
Class 5 the image and preimage of map, composition of maps, product set Details will be provided during each class session
Class 6 discussion session Details will be provided during each class session
Class 7 correspondence between sets, indexed set Details will be provided during each class session
Class 8 discussion session Details will be provided during each class session
Class 9 binary relation, equivalence relation, equivalence class, quotient set Details will be provided during each class session
Class 10 discussion session Details will be provided during each class session
Class 11 the cardinality of set, relation between cardinality, countable set Details will be provided during each class session
Class 12 discussion session Details will be provided during each class session
Class 13 cardinality of the continuum, uncountable set, cardinality of power set Details will be provided during each class session
Class 14 discussion session Details will be provided during each class session

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

### Textbook(s)

「集合と位相」内田伏一著　裳華房 (1986/2020年)

### Reference books, course materials, etc.

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