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.
Intercultural skills | Communication skills | ✔ Specialist 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 | Definition and examples of sets | Understand the contents covered by the lecture. |
Class 2 | Several operations on sets | Understand the contents covered by the lecture. |
Class 3 | mappings and their properties | Understand the contents covered by the lecture. |
Class 4 | family of sets indexed by a set, the axiom of choice | Understand the contents covered by the lecture. |
Class 5 | operations on sets and mappings, equivalence relations and quotient sets | Understand the contents covered by the lecture. |
Class 6 | the definition of cardinality of sets | Understand the contents covered by the lecture. |
Class 7 | Comparison of the cardinarity | Understand the contents covered by the lecture. |
Class 8 | counable sets | Understand the contents covered by the lecture. |
Class 9 | The cardinarity of the continuum and uncountable sets | Understand the contents covered by the lecture. |
Class 10 | orderrings | Understand the contents covered by the lecture. |
Class 11 | linearly ordered set | Understand the contents covered by the lecture. |
Class 12 | Zorn's lemma | Understand the contents covered by the lecture. |
Class 13 | Euclidean space | Understand the contents covered by the lecture. |
Class 14 | topological properties of Euclidean space | Understand the contents covered by the lecture. |
Class 15 | metric spaces | Understand the contents covered by the lecture. |
"Set and Topological Space (in Japanese)" by Shigeyuki Morita published by Aasakura Shoten.
Not specified in particular.
By scores of report (or the midterm examination) and the final examination.
The students are encouraged to take "MCS.T202：Exercises in Set and Topology I", simultaneously.