MA651 Topology 1

Instructor: Nikolay S. Strigul
Office Hours: by appointment

Lectures: Tuesdays 06:15-08:45 pm, Lieb Building 120
Homework: There will be weekly homework assignments due every two weeks.
Exams, Quizzes: There will be Midterm and Final Exams. Every class will start with a quiz covering definitions and theorems.


Course program Course program (PDF)

General comments:

MA651 is a graduate course in general topology. The major goal of this course is to cover basic topological ideas such as topological spaces and their products, continuous maps, compactness, connectedness, and metrization. However, some additional materials will be also covered in depth: for example, the two first lectures will be devoted to set theory. A lecture about real numbers will demonstrate how topological and algebraic ideas relate. An additional two lectures illustrating how topology contributes to functional analysis and dynamical systems will be given provided that the mandatory material is covered.

This course is targeted at graduate students in pure and applied mathematics. However MA651 is close to a self-contained course. Advanced undergraduate students are welcome if they are ready. To illustrate what the word "ready" means I offer a citation from the introduction to the excellent book by Marc Zamansky: "However, in mathematics, as in many other disciplines, although previous knowledge may be unnecessary, a certain attitude of mind is essential." Then, although no official prerequisites are listed, students should be able to operate with abstract ideas.

Textbooks (students should buy only the first textbook by James Munkres)

1) Topology by James Munkres, 2nd ed.
2) Topology by James Dugundji
3) Linear Algebra and Analysis by Marc Zamansky

Some materials will be taken from the other books:

1) Fundamentals of General Topology : Problems and Exercises by A.V. Arkhangel'skii, V.I. Ponomarev
2) Introduction to Set Theory and General Topology by P.S. Aleksandrof
3) Topology by K. Kuratowski v.1
4) Elements of Mathematics: General Topology by Nicolas Bourbaki
5) Introduction to Set Theory by K. Hrbacek and T. Jech
6) Introduction to Real Analysis by A.N. Kolmogorov and S.V. Fomin

Course program:

Jan 17. Lecture 1. - Elementary set theory.

1. Symbolic logic notation.
2. Sets.
3. Boolean algebra.
4. Cartesian product.
5. Families of sets.
6. Power set.
7. Functions, or maps.
8. Binary relations. Equivalence relation.
9. Axiomatics.
Lecture 1.pdf Homework 1.pdf Quiz questions 1.pdf

Jan 24. Lecture 2. - Ordinals and cardinals. Well-ordered sets.

10. Ordering.
11. Zorn's Lemma, Zermelo's Theorem and Axiom of Choice.
12. Ordinals.
13. The concept of ordinal numbers.
14. Comparison of ordinal numbers.
15. Transfinite induction.
16. Cardinality of sets.
17. Finite, countable and uncountable sets.
18. General cartesian products.
Lecture 2.pdf Homework 2.pdf Quiz questions 2.pdf Quiz 1.pdf

Jan 31. Lecture 3. - Topological spaces.

19. Fundamental families."Local" definition of topology.
20. "Global" definition of topology.
21. Basis for a given topology.
22. Topologizing of sets.
23. Open and closed sets.
24. Induced topology.
Lecture 3.pdf Homework 3.pdf Quiz questions 3.pdf Quiz 2.pdf

Feb 7. Lecture 4. - Continuity. Homeomorphisms. Limits.

25. Continuous maps.
26. Open maps and closed maps.
27. Homeomorphism.
28. Continuity from a "local" viewpoint.
        28.1 The concept of a filter.
        28.2 Limits in topological spaces.
        28.3 Images of limits, sequences.
        28.4 "Local" definition of a continuous map.
Lecture 4.pdf Homework 4.pdf Quiz questions 4.pdf Quiz 3.pdf

Feb 14. Lecture 5. - Cartesian product topology. Connectedness.

29. Cartesian product topology.
30. Slices in cartesian product topology.
31. Connectedness.
32. Application to real valued functions.
33. Components.
34. Local connectedness.
35. Path-connectedness.
Lecture 5.pdf Homework 5.pdf Quiz questions 5.pdf Quiz 4.pdf

Feb 21. No lecture (Stevens is on Monday schedule ) - takehome midterm exam.

        Midterm Exam.pdf

Feb 28. Lecture 6. - Separation axioms.

36. Separation axioms.
37. Hausdorff spaces.
38. Regular spaces.
39. Normal spaces.
40. Urysohn's characterization of normality.
41. Tietze's characterization of normality.
Lecture 6.pdf Homework 6.pdf Quiz questions 6.pdf Quiz 5.pdf

Mar 7. Lecture 7. - Real numbers.

42. Algebraic laws.
43. The set of rational numbers.
        43.1 The set Z of integers.
        43.2 Definitions and properties of the set Q of rationals.
        43.3 Topology on Q.
44. The construction of R and its fundamental properties.
        44.1 Definition of R.
        44.2 Addition, order, absolute value in R.
        44.3 The field R.
        44.3 The topology on R. The two fundamental properties.
45. The real line.
Lecture 7.pdf Homework 7.pdf Quiz questions 7.pdf Quiz 6.pdf

Mar 14. Spring recess. No lecture.

Mar 21. Lecture 8. - Compactness.

46. Compactness.
47. Compact subsets in R.
48. Compactness as a topological invariant.
49. Separation properties of compact spaces.
50. Tychonov Theorem.
51. Alexandroff compactification.
Lecture 8.pdf Homework 8.pdf Quiz questions 8.pdf Quiz 7.pdf

Mar 28. Lecture 9. - Compactness 2

52. Local compactness.
53. Countable compactness.
54. Sequential compactness.
55. 1o countable spaces.
56. 2o countable, separable and Lindelöf spaces.
Lecture 9.pdf Homework 9.pdf Quiz questions 9.pdf Quiz 8.pdf

Apr 4. Lecture 10. - Metric spaces 1

57. Metrics on sets.
58. Topology induced by a metric.
59. Equivalent metrics. Isometries.
60. Continuity of distance.
61. Convergent sequences.
62. Compactness and covering theorems in metric spaces.
Lecture 10.pdf Homework 10.pdf Quiz questions 10.pdf Quiz 9.pdf

Apr 11. Lecture 11. - Metric spaces 2

63. Complete metric spaces.
        63.1 Cauchy sequences.
        63.2 Complete metrics and complete spaces.
64. The Baire property of complete metric spaces.
65. Completion of a metric space.
Lecture 11.pdf Homework 11.pdf Quiz questions 11.pdf Quiz 10.pdf

Apr 18. Lecture 12. - Mapping in metric spaces.

66. Uniform continuity.
67. Extension by continuity.
68. Contraction mapping.
        68.1 The fixed point theorem.
        63.2 Contraction mapping and differential equations.
Lecture 12.pdf Homework 12.pdf Quiz questions 12.pdf Quiz 11.pdf

Apr 25. Lecture 13. - Review of homework assignments.

May 2. Lecture 14. - Review of the course. Application of topology to dynamical systems.

May 9. Final exam.