2021　Set and Topology I

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Mathematics
Instructor(s)
Kalman Tamas
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Tue3-4(H102)  Tue3-4(W931)
Group
-
Course number
ZUA.B201
Credits
2
2021
Offered quarter
1-2Q
Syllabus updated
2021/3/19
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 and ordered set, Euclidean space and general metric space. 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. Then we introduce the equivalence of sets, and learn the notion of cardinality. After introducing the basic notions of ordered set, well-ordered set, and inductive set, some applications of these concepts will be provided. We also introduce Euclidean space and learn that the continuity of maps between Euclidean spaces can be simply rephrased by making use of open sets. Finally, we discuss the notion of general metric space, and learn that the continuity of maps between them may also be simply described using open sets. We strongly recommend to take this course with "Exercises in Set and Topology".
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 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

Standard lecture course

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 De Morgan's law, distributive law, mapping between sets Details will be provided during each class session
Class 3 the image and preimage of map, composition of maps, product set Details will be provided during each class session
Class 4 correspondence between sets, indexed set Details will be provided during each class session
Class 5 binary relation, equivalence relation, equivalence class, quotient set Details will be provided during each class session
Class 6 the cardinality of set, relation between cardinality, countable set Details will be provided during each class session
Class 7 cardinality of the continuum, uncountable set, cardinality of power set Details will be provided during each class session
Class 8 order, total order, well-ordered set and their basic properties Details will be provided during each class session
Class 9 inductive set, Zorn's lemma Details will be provided during each class session
Class 10 ordinal number, comparison of cardinality Details will be provided during each class session
Class 11 Equivalence between the well-ordering theorem, Zorn’s lemma and the axiom of choice Details will be provided during each class session
Class 12 Application of Zorn's lemma Details will be provided during each class session
Class 13 Euclidean space, metric space, open set and closed set Details will be provided during each class session
Class 14 basic concepts on metric spaces 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.