Math 445: Introduction to Topology I

David Dumas

University of Illinois at Chicago
Spring 2019

Textbook	Topology, 2ed by James R. Munkres. (Prentice Hall, 2000) UIC library / Google books / Amazon
Lectures	MWF at 1pm in 309 Taft Hall
Instructor email	david@dumas.io
Instructor office hours	MWF 11am May 3 office hours end at 11:25am
CRN	39509 (undergraduate), 39510 (graduate)

About the course

Math 445 provides an introduction to topology, which is the field of mathematics concerned with a formalization of the notion of "shape". Most of the course will focus on the area within topology known as point set topology. We will define topological spaces and discuss some important examples, such as metric spaces. We will study a variety of properties of topological and metric spaces, including compactness and connectedness. We will also discuss general methods for constructing new topological spaces from existing ones, such as products, quotients, and subspaces.

We will cover chapters 2-4 in the textbook and selected topics from chapters 5-8.

Documents

• Course syllabus
• Midterm exam
• Midterm exam solutions
• Final exam problems 1 and 2

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 6	20%
Final Exam	Mon May 6	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 309 Taft Hall).

• Homework 0 due Friday January 18: Read the syllabus
• Homework 1, due Monday January 28
• Homework 2, due Monday February 4
• Homework 3, due Monday February 11:
  Problems 17.11, 17.13, 18.2, 18.4, 18.7, and 18.10 from Munkres
• Homework 4, due Monday February 18
• Homework 5, due Monday February 25
• Homework 6, due Monday March 4:
  Problems 21.7, 21.9, 22.4, 23.2, 23.4, and 23.8^* from Munkres
  * Hint: Consider the set of bounded sequences (a subset of R^ω)
• Homework 7, due Monday March 11:
  Problems 24.1ab, 24.3^*, 24.8abc, 24.10
  * Hint: Consider the function g(x) = f(x)-x. It is continuous and if f has no fixed point then g is nowhere zero.
• Homework 8, due Monday March 18:
  Problems 25.1, 25.2ab, 26.2a, 26.3, 26.4, 26.6
• Homework 9, due Wednesday April 3
• Homework 10, due Monday April 8:
  Problems 28.1, 28.6, 29.8^*
  * Section 29 has two lists of exercises: The regular exercises on page 186, and a collection of supplementary exercises about nets (a topic we are not covering) that starts on the next page. Both lists have an exercise #8. You should do the one on page 186, about the one-point compactification of the positive integers.
• Homework 11, due Monday April 15:
  Problems 30.2, 30.9, 30.13, 30.16a, 31.1, 31.2, 31.5
• Homework 12, due Monday April 22:
  Problems 31.7^*, 32.1, 32.2, 32.5, 33.2, 34.3
  * Reminder: A function f:X→Y is closed if f(C) is closed whenever C is a closed subset of X.
• Homework 13, due Monday April 29
• Suggested exercises for week 15 (not collected)::
  46.1, 46.3, 46.5, 47.1, 47.5, 48.2, 48.3

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 (Mon Jan 14): Topologies and topological spaces (§13)
• Lecture 2 (Wed Jan 16): The topology determined by a basis (§13)
• Lecture 3 (Fri Jan 18): Recognizing a basis of a topology, comparing topologies (§13)
• Lecture 4 (Wed Jan 23): The product topology (§15)
• Lecture 5 (Fri Jan 25): The subspace topology (§16)
• Lecture 6 (Mon Jan 28): Closed sets and limit points
• Lecture 7 (Wed Jan 30): (class canceled due to weather)
• Lecture 8 (Fri Feb 1): T₁ and T₂ spaces (§17)
• Lecture 9 (Mon Feb 4): Continuity (§18)
• Lecture 10 (Wed Feb 6): Homeomorphisms, embeddings, examples of continuous functions (§18)
• Lecture 11 (Fri Feb 8): Criteria for continuity, pasting lemma (§18)
• Lecture 12 (Mon Feb 11): Box and product topologies (§19)
• Lecture 13 (Wed Feb 13): Box and product topologies (continued) (§19)
• Lecture 14 (Fri Feb 15): The metric topology; metric space examples (§20)
• Lecture 15 (Mon Feb 18): Metrics on R^J (§20)
• Lecture 16 (Wed Feb 20): ε-δ continuity (§21)
• Lecture 17 (Fri Feb 22): Pointwise and uniform convergence; uniform limit theorem (§21)
• Lecture 18 (Mon Feb 25): Examples of Hausdorff, non-metrizable spaces (§20)
• Lecture 19 (Wed Feb 27): The quotient topology (§22)
• Lecture 20 (Fri Mar 1): Connectedness (§23), intervals in R are connected (§24)
• Lecture 21 (Mon Mar 4): Connectedness of closures and of R^ω
• Lecture 22 (Wed Mar 6): Path connectedness (§24)
• Lecture 23 (Fri Mar 8): A space that is connected but not path connected (§24)
• Lecture 24 (Mon Mar 11): Components and local connectedness (§25)
• Lecture 25 (Wed Mar 13): Compactness (§26)
• Lecture 26 (Fri Mar 15): Properties of compactness (§26)
• Lecture 27 (Mon Mar 18): Compactness of finite products and of closed intervals in R (§26-27)
• Lecture 28 (Wed Mar 20): Distance to a set in a metric space; The Lebesgue number lemma (§27)
• Lecture 29 (Fri Mar 22): Uniform continuity theorem; Sequential compactness (§28)
• Lecture 30 (Mon Apr 1): Sequential compactness vs. compactness for metrizable spaces (§28)
• Lecture 31 (Wed Apr 3): The one-point compactification (§29)
• Lecture 32 (Fri Apr 5): First and second countability, Lindelöf, and separability (§30)
• Lecture 33 (Mon Apr 8): The separation axioms: Hausdorff, regular, and normal spaces (§31)
• Lecture 34 (Wed Apr 10): Normal spaces (§32)
• Lecture 35 (Fri Apr 12): Normal spaces (continued), Urysohn's lemma (§33)
• Lecture 36 (Mon Apr 15): Proof of Urysohn's lemma (§34)
• Lecture 37 (Wed Apr 17): Urysohn metrization theorem (§34), Completeness (§43)
• Lecture 38 (Fri Apr 19): Complete metric spaces (§43)
• Lecture 39 (Mon Apr 22): Function spaces (§43)
• Lecture 40 (Wed Apr 24): Equicontinuity (§45)
• Lecture 41 (Fri Apr 26): The Arzela-Ascoli Theorem for X→Rⁿ with X compact (§45)
• Lecture 42 (Mon Apr 29): Compact convergence (§46)
• Lecture 43 (Wed May 1): The general Arzela-Ascoli Theorem (§47)
• Lecture 44 (Fri May 3): The Baire Category Theorem (§48)

Reading assignments

• Jan 14: Chapter 2 (and skim Chapter 1)
• Feb 25: Chapter 3
• Mar 30: Chapter 4
• Apr 17: Chapter 7
• Apr 29: Section 48

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.

Online resources

• Spaces that are connected but not path-connected, lecture notes by Keith Conrad
• π-base
• Notes on the Tychonoff theorem by Pete L. Clark