Topology

Instructor: Xiaolong Li

Lectures: MWF 10:00 - 10:50 in MSTB 122

Office Hours: MWF 11:00 -11:50 in RW 510W.

Email: xiaolol1@uci.edu

Discussion Sections: TuTh 11:00 -11:50 in MSTB 122

Teaching Assistant: Hongyi (Henry) Sheng

Office Hours: TBA

Email: hsheng2@uci.edu

Course Information:

Textbook: Introduction to Topology, by T. Gamelin and R. Greene.

(We cover at least the first two chapters. I wish to cover part of Chapter 3.).

Grading Policy:

Final Exam: (45%)

June 10, Monday, 10:30 -12:30 pm

Midterm: (35%)

May 3, in class, covers Chapter 1.

Homework: (20%) (Collected in discussion sections on Thursdays)

Homework 1: (due Apr 11)

Section 1.1: Exercises 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

Homework 2: (due Apr 18)

Section 1.2: Exercises 3, 4, 5, 6, 7, 8Section 1.3: Exercise 9Section 1.4: Exercises 1, 2, 3, 4

Homework 3:(due Apr 25)

Section 1.4: Exercise 7 Section 1.5: Exercises 1, 2, 3, 5, 7Additional Problems:1. Prove that a subset E of R^n (n-dimensional Euclidean space) is totally bounded if and only it E is bounded (Lemma 5.3 in the textbook).2.Prove that a subspace of a totally bounded metric space is totally bounded (Lemma 5.4 in the textbook). 3. Prove the Heine-Borel Theorem (Theorem 5.5 in the textbook).
A hint for exercise 7 in section 1.5: S is finite implies B(S) is separable: If S is finite, let C be the subset of B(S) consisting of rational-valued functions. Then C is a countable dense subset of B(S) (verify this). B(s) is separable implies S is finite: Suppose S is not finite, then the power set of S(all the subsets of S) is uncountable. By the hint in the exercise, we have uncountably many disjoint balls of radius 1/2. Let A be a countable dense subset of B(S). Since a dense subset intersects every ball, A must contain at least one element from each of the uncountably many disjoint balls. Thus A must be uncountable. This is a contradiction.

Homework 4:(due May 9)

Section 1.6: Exercises 1, 6, 7, 8Section 2.1: Exercises 2, 3, 4, 6, 7, 8, 9, 12Section 2.2: Exercises 2, 4

Homework 5:(due May 16)

Section 2.3: Exercises 2, 4, 5, 6, 7, 13Section 2.4: Exercises 1, 4, 6
Additional Problems: 1. Prove Lindelof's Theorem: Let X be a second-countable topological space. Then every open cover of X has a countable subcover. 2. Prove that a second-countable topological space is separable.

Homework 6:(due May 23)

Section 2.5: Exercises 1, 3, 4, 6, 7, 8Section 2.6: Exercises 1, 2 ,3, 4, 5, 6 Additional Problem: Prove Theorem 6.7 on Page 81. (Read Exercises 7, 8, 9, 10, 11 in Section 2.6)

Homework 7:(due May 30)

Section 2.8: Exercises 3, 4, 5, 6, 9Section 2.9: Exercises 1, 4, 6, 7Section 2.10: Exercises 2, 4

Homework 8:(due June 6)

Section 2.13: Exercises 1, 4, 6, 8Review problems on the midterm

Calendar:

Week 1:

Apr 1: Section 1.1 Apr 3: Section 1.1Apr 5: Section 1.2

Week 2:

Apr 8: Section 1.2Apr 10: Section 1.2Apr 12: Section 1.3, 1.4

Week 3:

Apr 15: Section 1.4Apr 17: Section 1.5Apr 19:Section 1.5

Week 4:

Apr 22: Section 1.6Apr 24: Section 2.1Apr 26: Section 2.1

Week 5:

Apr 29: Section 2.2 May 1:ReviewMay 3: Midterm

Week 6:

May 6: Section 2.3May 8: Section 2.4May 10: Section 2.5

Week 7:

May 13: Section 2.5May 15: Section 2.6May 17: Section 2.8

Week 8:

May 20: Section 2.9May 22: Section 2.10May 24: Section 2.10, 2.13

Week 9:

May 27: Memorial DayMay 29: Section 3.2May 31: Section 3.3

Week 10:

Jun 3: Section 3.4 Jun 5: Section 3.5Jun 7: Review

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