Math 551 (Fall 2017)
Elementary Topology
Elementary Topology
Announcements
Take-home final is available here (Click "Final Exam" to download). It is due at 9:25pm on December 19, 2017. Please submit your answers by email at luwang@math.wisc.edu.
Office Hours of the week of Nov. 13: Monday, 11am-Noon and 4:30pm-6:30pm, in 809 Van Vleck.
HW#10 was post and is due Monday, Nov. 20.
HW#6 was post and is due Friday, Oct. 20.
HW#5 is due Monday, Oct. 16.
Office Hours of the week of Oct. 9: Monday, 11am-Noon and 4:30pm-6:30pm, in 809 Van Vleck.
Welcome to the class! The first lecture is at 12:05pm on Wednesday, Sep. 6, in B135 Van Vleck.
Lectures
Time: 12:05pm-12:55pm, on Mondays, Wednesdays, and Fridays
Location: B135 Van Vleck Hall
Instructor
Lu Wang
Office: 809 Van Vleck Hall
Email: luwang@math.wisc.edu
Office Hours
Time: 11am-Noon, on Mondays and Wednesdays
Location: 809 Van Vleck Hall
Textbook
Title: Topology (2nd Edition)
Author: James Munkres
Publisher: PEARSON
ISBN 13: 9780131816299
Syllabus
Elementary Set Theory: sets, Cartesian products, equivalence relations, countability, the real line
Basic Topology: metric spaces, topological spaces, basis, continuity, connectedness, compactness
Homework
There will be weekly homework assignments mostly selected from the textbook. Assignments are usually posted here on Fridays and due in the beginning of the lecture on the following Friday. No late homework will be accepted, but the lowest two scores will be dropped when calculating the course grade.
HW#1 (due Sep. 15): Section 1: #1, 2, 8, 10
HW#2 (due Sep. 22): Section 2: #1, 4, 5; Section 3: #4, 9; Section 5: #1
HW#3 (due Sep. 29): Section 6: #2, 3, 5, 6; Section 7: #1, 2, 3. For Honor Students: Section 6: #7
HW#4 (due Oct. 6): Section 13: #1, 3, 4, 5. For Honor Students: Section 13: #7
HW#5 (due Oct. 16): Section 16: #1, 3, 4, 6; Section 17: #6, 8(a)(b). For Honor Students: Section 16: #9, 10; Section 17: #8(c)
HW#6 (due Oct. 20): Section 17: #9, 11, 12, 13. For Honor Students: Section 17: #19
HW#7 (due Oct. 27): Section 18: #3, 4, 5, 6, 10, 11, 12
HW#8 (due Nov. 3): Section 19: #1, 2, 3; Section 20: #1, 3
HW#9 (due Nov. 10): Section 21: #2, 3, 6, 8
HW#10 (due Nov. 20): Section 22: # 2, 3; Section 23: #1, 2
HW#11 (due Dec. 1): Section 23: #5, 9; Section 24: #1, 2, 3; Section 26: #3, 4
HW#12 (not turn in): Section 26: #5, 6, 7, 8; Section 27: #2, 6
Exams
There will be two close-book (i.e., no textbook, notes, homework, etc. are allowed), in-class midterm exams and one take-home final. Make-up exams will be provided only with the instructor's consent, and only in case of illness, academic conflicts, or family emergency. The tentative exam times are listed here.
Midterm I: Oct. 11, in class (Chapter 1: Sections 1--7; Chapter 2: Sections 12--15)
Midterm II: Nov. 15, in class (Chapter 2: Sections 16--21)
Final: due Dec. 19, take-home
Grading
Homework: 20%
Midterm I: 25%
Midterm II: 25%
Final: 30%
Letter grades for the course will be roughly assigned according to the following standard:
A: 90+
AB: 86+
B: 77+
BC: 72+
C: 60+
D: 48+
F: 0+
Lecture Schedule
The following lecture schedule will be updated during the semester.
Week of Sep. 6
Introduction
Logic and elementary set theory: Munkres 1-1
Week of Sep. 11
Functions, domain and range: 1-2
Injective, surjective, bijective functions: 1-2
Inverse image of sets in range of functions: 1-2
Relations and equivalence relations: 1-3
Order relations and dictionary relations: 1-3
Week of Sep. 18
Supremum and infimum: 1-3
Real numbers, well-ordering property and induction: 1-4
Finite and countable Cartesian product: 1-5
Finite sets: 1-6
Proof that cardinality of a finite set is well-defined: 1-6
Finite unions and finite cartesian products of finite sets are finite: 1-6
Week of Sep. 25
Infinite sets, countably infinite: 1-7
Countable union of countable sets is countable
Finite Cartesian product of countable sets is countable
{0,1}^\omega is uncountable
The set of real numbers is uncountable
The power set of X cannot be in bijection with X
Definition of a topology: 2-12
Discrete and trivial topologies
Week of Oct. 2
Basis for a topology: 2-13
Subbasis for a topology
The Order Topology: 2-14
The Product Topology: 2-15
Week of Oct. 9
The Subspace Topology: 2-16
Midterm I on Oct. 11
Closed sets: 2-17
Closure and interior of a set
Week of Oct. 16
Limit points: 2-17
Hausdorff spaces
Continuous functions: 2-18
Homeomorphisms and embeddings
Week of Oct. 23
Rules for continuous functions: 2-18
Pasting together continuous functions
Continuous maps into product spaces
Arbitrary Cartesian products: 2-19
Comparison of Box and Product Topologies
Metric topology: 2-20
Week of Oct. 30
Triangle Inequality for R^n: 2-20
Comparison of metric topologies, Euclidean vs. square metric
Continuity in metric spaces: 2-21
Theorem 21.1: epsilon-delta continuity of metric spaces
Theorem 21.3: Limit point definition of continuity
Week of Nov. 6
Theorem 21.6: Uniform Limit Theorem
Quotient Topology: 2-22
Week of Nov. 13
Continuous functions on quotient spaces: 2-22
Connectedness: 3-23
Midterm II on Nov. 15
Closure of connected set is connected: 3-23
Finite Cartesian product of connected spaces is connected
Week of Nov. 20
Image of connected set under continuous map is connected: 3-23
The real line is connected, and so are intervals and rays: 3-24
Intermediate Value Theorem
Path connectedness
The topologist's sine curve
No class on Nov. 24 due to thanksgiving recess
Week of Nov. 27
Compactness: 3-26
Closed subspace of compact space is compact
Compact subspace of Hausdorff space is closed
Image of compact set under continuous map is compact
The product of finitely many compact spaces is compact
The Tube Lemma
Subset of R^n is compact iff closed and bounded: 3-27
Week of Dec. 4
Compact subspace of the real line: 3-27
Extreme Value Theorem
Lebesgue Number Lemma
Theorem 27.6: Uniform Continuity Theorem
Week of Dec. 11
TBA