The objective of this course is to explain the fundamentals of set theory and topology. The main theme of this cousrse is set theory. In the first half of this course, lectures on sets, maps, axiom of chice, equivalence relations, and the cardinarity of sets are given. In the the last half of this course, orderd set, Zorn's lemma, and the topological properties of metric spaces are given. This course is aimed to connected to the course "Set and Topology II" in the third quarter.
(Theme)
The set and topology play an important role in mathematical and computing science. The objective of this course is to explain the fundamentals of set theory,
and topological properties of metric spaces as an introduction of topology.
(The goal)
The students are expected to understand the fundamentals of mathematical methods to handle the concept of sets, maps, and metric spaces appeared in mathematical and computing science and also to be able to apply them to practical problems.
set, mapping, family of sets indexed by a set, the axiom of choice, equivalence relations, quotient set, cardinality of sets, countabe set, uncountable set, order, linearly ordered set, Zorn's lemma, Euclidean space, metric space.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
The lectures provide the fundamentals of set theory and an introduction of topology of metric spaces. The students are strongly encouraged to register for "MCS.T202：Exercises in Set and Topology I" simultaneously which offers the recitation session for this course.
Course schedule | Required learning | |
---|---|---|
Class 1 | Sets and their operations | Understand the contents covered by the lecture. |
Class 2 | mappings and their properties | Understand the contents covered by the lecture. |
Class 3 | Cartesian products and graphs of mappings | Understand the contents covered by the lecture. |
Class 4 | Indexed families of sets | Understand the contents covered by the lecture. |
Class 5 | Equivalence relations and quotient sets | Understand the contents covered by the lecture. |
Class 6 | Definition of cardinality of sets and comparison of the cardinarity | Understand the contents covered by the lecture. |
Class 7 | Countable sets, the cardinarity of the continuum, and uncountable sets | Understand the contents covered by the lecture. |
Class 8 | Operations on cardinalities and continuum hypothesis | Understand the contents covered by the lecture. |
Class 9 | orderings and well-ordered sets | Understand the contents covered by the lecture. |
Class 10 | Zorn's lemma | Understand the contents covered by the lecture. |
Class 11 | Ordinal numbers | Understand the contents covered by the lecture. |
Class 12 | Applications of Zorn's lemma and well-ordering theorem | Understand the contents covered by the lecture. |
Class 13 | Euclidean spaces | Understand the contents covered by the lecture. |
Class 14 | Metric spaces | Understand the contents covered by the lecture. |
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.
"Set and Topological Space (in Japanese)" by Shigeyuki Morita published by Aasakura Shoten.
Specified by the lecturer.
Will be based on exercise and/or report. If you register for "MCS.T202:Exercises in Set and Topology I", its score will be counted as a part of contributions also. Details will be announced in the first lecture.
The students are strongly encouraged to take "MCS.T202：Exercises in Set and Topology I", simultaneously.