基本素養 Basic Literacy

核心能力 Competence

掌握連續與逼進的能力
Ability to handle continuity phenomena and approximations.
處理數字及符號運算的能力
Ability to operate with symbols and digits.
透視形像與空間的能力
Ability to visualize space, shape, and images.
利用計算機處理數學的能力
Ability of using computer to do mathematics.
應用隨機理論的能力
Ability to apply stochastic mathematics.
處理大量數據的能力
Ability to handle large amount of data.
處理離散型數學的能力
Ability to do mathematics relating to discrete models.
獨立研究的能力
Ability to do independent studies of mathematics.
使用計算機的能力
Abilities to use computer as a tool.

課程概述 Course Description

拓樸學研究幾何物體(拓樸空間)和它的分類問題。它探討物體在連續形變下不變的性質,譬如:正方形和圓形在拓樸上是一樣的。這門課將從拓樸空間和連續函數的定義出發,接著引進連通性、緊緻性、緊緻化、商空間、正規空間等基本概念。我們也探討Urysohn 引理和Tychonoff 定理,以及淺談基本群及覆疊空間。
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