## 核心能力 Competence

Ability to handle continuity phenomena and approximations.

Ability to operate with symbols and digits.

Ability to visualize space, shape, and images.

Ability of using computer to do mathematics.

Ability to apply stochastic mathematics.

Ability to handle large amount of data.

Ability to do mathematics relating to discrete models.

Ability to do independent studies of mathematics.

Abilities to use computer as a tool.

## 課程概述 Course Description

In topology, we study geometric objects (topological spaces) and their classification. We are interested in properties of objects that are preserved when one object is continuously deformed into another. For instance, a square and a circle are topologically the same. In this course, we will begin by defining what topological spaces and continuous functions are. We will then discuss connectedness, compactness, compactification, quotient spaces, normal spaces, Urysohn’s Lemma, and Tychonoff’s Theorem, and give an elementary introduction to fundamental groups and covering spaces.

## 課程學習目標 Course Objectives

• 度量空間之基本知識
• 拓樸空間之緊緻、連通、分離概念
• 代數拓樸入門
• ## 課程進度 Progress Description

進度說明 Progress Description
1Introduction to metric spaces
2Open and closed sets in metric spaces, continuous functions
3Convergence and completeness, the real line
4Compact metric spaces
5Heine-Borel Theorem and equivalent forms
6Midterm 1
7Introduction to topological spaces
8Generalization from metric spaces
9Base of topology
10Separation Axioms
11Compactness
12Midterm 2
13Connectedness
14More topics in point set topology
15The fundamental groups
16Covering spaces
17Other topics in algebraic topology
18Final Exam
以上每週進度教師可依上課情況做適度調整。The schedule may be subject to change.

## 有關課程其他調查 Other Surveys of Courses

1.本課程是否規劃業界教師參與教學或演講? 否
Is there any industry specialist invited in this course? How many times? No
2.本課程是否規劃含校外實習(並非參訪)? 否
Are there any internships involved in the course? How many hours? No
3.本課程是否可歸認為學術倫理課程? 否
Is this course recognized as an academic ethics course? In the course how many hours are regarding academic ethics topics? No
4.本課程是否屬進入社區實踐課程? 否
Is this course recognized as a Community engagement and Service learning course? Which community will be engaged? No