拓扑学

 

本课程内容和进度大致如下:

1 Sets

1.1 Sets, intersections of sets, unions of sets, Cartesian products of sets (§1,§5).
1.2 Maps, domain and range, image and inverse image, injective maps, surjective maps, identity map, one-to-one correspondence, category of sets (§2).
1.3 Equivalence relations, equivalence classes, the set of equivalence classes and quotient map, axiom of choice (§3,§9).
1.4 Equivalence classes on the category of sets, finite sets, countable sets, uncountable sets (§3,§6,§7).
1.5 Order relations, simply ordered sets, partially ordered sets, upper bounds, maximal element, Zorn's Lemma (§3,§11).


2 Topological spaces

2.1 Review of limits and continuity of functions in one real variable.
2.2 Metric spaces and limit in metric spaces (§20).
2.3 Continuity of maps between metric space in terms of open balls (§21).
2.4 Topologies on a set and topological spaces, open sets, trivial topology and discrete topology, finer or coarser topology (§12).
2.5 Basis of a tolopogy, topology generated by a basis, topology as unions of elements in the basis, comparison of topologies on the level of bases (§13).
2.6 Continuous maps, checking the continuity of a map on a basis, continuity of composition of two continuous maps, continuity of identity map, category of toplogical spaces, homeomorphisms as equivalence relation on the category of topological spaces (§18).
2.7 Subspace topology, basis of the subspace topology, open sets in open subspaces, continuity of the inclusion map, subspace topology as the minimal topology that makes the inclusion map continuous, categorical description of the subspace (§16).
2.8 Quotient topology, basis of the quotient topology, open sets in open subspaces, continuity of the quotient map, quotient topology as the maximal topology that makes the quotient map continuous, categorical description of the quotient space (§22).
2.9 Product topology on cartesion product of two topological spaces, continuity of the projection maps, product topology as the topology that makes the two projections continuous, categorical description of the product space, comparison with the disjoint union (§15).
2.10 Gluing of continuous maps on open subsets, relative product (§18).
2.11 Closed sets, properties of closed sets, topology in terms of closed sets, closure and interior of a set, limit points (§17).

3 Connectedness

3.1 Connected spaces (§23).
3.2 Path connected spaces (§24).
3.3 Connected components, locally connected spaces (§25).

4 Compactness

4.1 Compact spaces (§26).
4.2 Compact subspaces of a Euclidean space (§27).
4.3 Limit point compactness (§28).
4.4 Local compactness (§29).

5 Countability (§30)

5.1 First countable spaces
5.2 Second countable spaces
5.3 Lindlof spaces
5.4 Separable space


6 Separation axioms(§31, §32)

6.1 T_1 spaces
6.2 T_2 spaces (Hausdorff spaces)
6.3 Regular spaces(Brief introduction)
6.4 Normal spaces(Brief introduction)

7 Metrization theorems(§34) (Brief introduction)

8 Fundamental groups and covering spaces(Chapter 9, Chapter 10, Chapter13)

 

课程笔记正在整理中。

拓扑学课程介绍

上面的图是荷兰著名版画艺术家M.C.Escher的一幅作品。在这里有关于他的更多介绍。

这幅版画里有什么样的拓扑学内容呢?按这里可以看到更大的图。


关于教材:

    我们使用的教材是机械工业出版社引进的英文原版教材J. R. Munkres, Topology, Second Edition. 这是2000年的第二版,此书1975年第一版Topology: A First Course曾于1985年由罗嵩龄、许依群、徐定宥、熊金城译成中文,1987年由科学出版社出版, 中文名为《拓扑学基本教程》。以下摘自他们的《译后记》:

    "一九七九年,北京大学江泽涵、姜伯驹教授向我们介绍了一本受到美国数学界推崇的书--Topology: A First Course (J. R. Munkres)。后来读到这本书,感到对于教学及研究确实大有裨益。作者通过创造性的工作,把拓扑学中艰深的概念、繁复的证明,安排得井然有序,由浅入深。对那些极度抽象的理论,作者采用生动的比喻把"背景"叙述得直观形象,起到了"点化"初学者心中模糊认识的作用, 不少精辟的论述,使人茅塞顿开,大有豁然贯通之感。

    正如"序言"所说, 本书兼顾了点集拓扑与代数拓扑两部分内容,考虑到一学期与两学期的不同教学时数, 许多章节相对独立成篇, 第一部分的"附加习题"更指明了深入研究的方向, 这种独具匠心的编排使高等学校数学系师生、研究生乃至自学者,有着极大的选择自由。"

    他们的这两段话现在看起来我还是非常的认同。第二版与第一版比较起来, 有了很大的扩充,主要是在代数拓扑部分。这反映了当前一种趋势:减少点集拓扑的部分,增加代数拓扑的介绍, 如基本群与覆盖空间的内容、曲面的分类及其基本群等。这部分内容是几何与拓扑中相当基本的内容, 经常用到,但很少会有专门的课程来介绍。

    我个人认为不认真学习点集拓扑的内容的话学生很难具备学习现代数学所需的语言、工具和抽象思维的能力, 但过分深究点集拓扑部分的内容对学生形成良好的数学品位是有害的, 所以在学习点集拓扑的同时学习一些代数拓扑是非常高明的策略。 这正是我们所用教科书的特点: 着重介绍点集拓扑中最常用的概念和方法,同时介绍一点代数拓扑。

    对于知道了有中译本就不想读英文原版书的同学,我需要提醒你,如果你想做一个正式的、严肃的数学工作者的话,你将来阅读和写作的绝大部分文章是英文的。将英文数学书翻译成中文的做法正在绝迹。你现在不应该去读中文译本或中文参考书,因为你已经大学三年级了,应该是足够成熟可以适应各种各样的教材与教师了。


Last modified: Wednesday, December 1, 2004 14:45 PM