Math 445: Introduction to Topology I

David Dumas

University of Illinois at Chicago
Spring 2018

Dynamical Lakes of Wada, after Coudene and Plykin.

Textbook	Topology, 2ed by James R. Munkres. (Prentice Hall, 2000) UIC library / Google books / Amazon
Lectures	MWF at 1pm in 308 Taft Hall
Instructor email	david@dumas.io
Instructor office hours	Mon, Wed, and Fri 2-3pm in 503 SEO Exception: No office hours April 13
Grader	Charles Alley (calley2@uic.edu)
Grader office hours	Mon 9am-12noon in 430 SEO
CRN	39509 (undergraduate), 39510 (graduate)

Make-up lecture

The optional make-up lecture is scheduled for Thursday, May 3, 2018 in SEO 612.

Note that the location has changed from the one previously announced.

Final exam information

The final exam for Math 445 is scheduled for Monday, May 7, 1:00pm - 3:00pm in 308 Taft Hall.

On the exam you will be asked to complete any four problems from a list of six problems.

Every problem on the exam will require you to write a proof.

The exam will be cumulative, and may include problems about any material from the lectures or from the corresponding sections of the textbook (see the list of lecture topics and textbook sections below).

Documents

• Course Syllabus
• Midterm exam and midterm solutions

Calendar & Grading

This is a condensed summary of some important dates and the weights used to compute your course grade. The course syllabus is the definitive reference for course policies.

	Date/time	Course grade fraction
Homework	Most Mondays (see list below)	50%
In-class midterm	Wed Mar 7	20%
Final Exam	Mon May 7	30%

Homework

Problem sets and their due dates are listed below. Unless otherwise noted on the assignment, homework is due in class at the beginning of the lecture (i.e. at 1:00pm in 308 Taft Hall).

• Homework 0 due Friday January 19: Read the syllabus
• Homework 1 due Monday January 29
• Homework 2 due Monday February 5
• Homework 3 due Monday February 12
• Homework 4 due Monday February 19: Complete these problems from the textbook:
  18.6, 18.9, 18.11, 19.1, 19.3, 19.7, 19.8
• Homework 5 due Monday February 26
• Homework 6 due Monday March 5
• Homework 7 due Monday March 12: Complete these problems from the textbook:
  23.1, 23.2, 23.3, 23.4, 23.5, 23.7
• Homework 8 due Monday March 19
• Homework 9 due Monday April 2
• Homework 10 due Monday April 9
• Homework 11 due Monday April 16: Complete these problems from the textbook:
  31.6, 32.3, 33.2(a), 34.3, 34.4
• Homework 12 due Monday April 23
• Homework 13 due Monday April 30: Complete these problems from the textbook:
  43.5, 43.8, 45.2, 21.9, 45.4

Typeset solutions are not required. If writing solutions by hand, please make sure they are legible. Staple homework if it spans several sheets of paper. Write your name and the assignment number (e.g. "Homework 1") at the top of the first page.

Many of the homework problems are assigned directly from the primary textbook (Munkres, 2ed).

Lecture titles and textbook sections

• Lecture 1 (Wed Jan 17): Topologies and topological spaces (§13)
• Lecture 2 (Fri Jan 19): The topology determined by a basis (§13)
• Lecture 3 (Mon Jan 22): More about bases (§13)
• Lecture 4 (Wed Jan 24): The order topology (§14)
• Lecture 5 (Fri Jan 26): The product toplogy and the subspace topology (§15-16)
• Lecture 6 (Mon Jan 29): More on subspaces; Closure and interior (§16-17)
• Lecture 7 (Wed Jan 31): Neighborhoods and limit points(§17)
• Lecture 8 (Fri Feb 2): Convergence, Hausdorff spaces, and T₁ spaces (§17)
• Lecture 9 (Mon Feb 5): Continuous functions (§18)
• Lecture 10 (Wed Feb 7): Equivalent conditions for continuity (§18)
• Lecture 12 (Mon Feb 12): Constructing continuous functions (§18-19)
• Lecture 13 (Wed Feb 14): The product topology for arbitrary products (§19)
• Lecture 14 (Fri Feb 16): Metric spaces (§20)
• Lecture 15 (Mon Feb 19): The metric topology (§20)
• Lecture 16 (Wed Feb 21): Metrics on Rⁿ and R^ω (§20)
• Lecture 17 (Fri Feb 23): More on R^ω and R^J; ε-δ continuity in metric spaces (§20-21)
• Lecture 18 (Mon Feb 26): Sequences in metric spaces; Uniform convergence (§21)
• Lecture 19 (Wed Feb 28): The quotient topology (§22)
• Lecture 20 (Fri Mar 2): Properties of quotients and examples (§22)
• Lecture 21 (Mon Mar 5): Connectedness (§23)
• Lecture 22 (Wed Mar 7): Midterm exam
• Lecture 23 (Fri Mar 9): R is connected; Intermediate Value Theorem (§24)
• Lecture 24 (Mon Mar 12): Path connectivity; Components (§24-25)
• Lecture 25 (Wed Mar 14): Local connectivity; Compactness (§25-26)
• Lecture 26 (Fri Mar 16): Compactness (§26-27)
• Lecture 27 (Mon Mar 19): Compactness in R and Rⁿ (§27)
• Lecture 28 (Wed Mar 21): Compact metric spaces; Lebesgue numbers (§27)
• Lecture 29 (Fri Mar 23): Sequential compactness and compactness (§28)
• Lecture 30 (Mon Apr 2): Countability axioms (§30)
• Lecture 31 (Wed Apr 4): Separation axioms (§31)
• Lecture 32 (Fri Apr 6): Regular and normal spaces (§31-32)
• Lecture 33 (Mon Apr 9): The Urysohn Lemma (§33)
• Lecture 34 (Wed Apr 11): The Urysohn Metrization Theorem (§34)
• Lecture 35 (Fri Apr 13): Complete regularity (§34)
• Lecture 36 (Mon Apr 16): The Tychonoff Theorem (§37)
• Lecture 37 (Wed Apr 18): Complete metric spaces (§43)
• Lecture 38 (Fri Apr 20): Completeness of Rⁿ and of the uniform topology on Y^X (§43)
• Lecture 39 (Mon Apr 23): Spaces of continuous functions and bounded functions (§45)
• Lecture 40 (Wed Apr 25): Equicontinuity (§45)
• Lecture 41 (Fri Apr 27): Equicontinuity and total boundedness in function spaces Rⁿ (§45)
• Lecture 42 (Mon Apr 30): The Arzela-Ascoli Theorem for functions to Rⁿ (§45)
• Lecture 43 (Wed May 2): The compact-open topology and the compact convergence topology (§46)
• Optional make-up lecture (Thu May 3): The Lakes of Wada
• Lecture 44 (Fri May 4): The general Arzela-Ascoli Theorem (§47)

Supplementary texts

For students looking for another place to read about the topics from the course, I recommend:

• Topology by Klaus Jänich, Springer, 1984.
  • Chapters 1, 3, 4, 6, 8, and 10 contain material we will cover in math 445.
• Introduction to Topology, 2ed by Theodore Gamelin and Robert Greene, Dover, 1999.
  • This book is terse but clearly written. It begins by discussing the topology of metric spaces in some detail, introducing general topological spaces a bit later.
  • Chapters 1 and 2 contain material we will cover in math 445.
• Topology, by John G. Hocking and Gail S. Young. Addison-Wesley, 1961.
  • Has a discussion of the Lakes of Wada on pp143-145.

Online resources

• π-base
• Notes on the Tychonoff theorem by Pete L. Clark
• Lakes of Wada:
  • Image gallery by Y. Coudene
  • Pictures of Hyperbolic Dynamical Systems by Y. Coudene, Notices of the American Mathematical Society, 53, No. 1, 2006.
  • Description at Cut the knot (basically a summary of the original construction of Yoneyama)
  • Original paper of Yoneyama (1917), see page 60.
  • Source code used to generate the image on the syllabus and the images shown in class on May 3.