\( \DeclareMathOperator{\Cl}{Cl} \DeclareMathOperator{\Fr}{Fr} \newcommand{\R}{\mathbb{R}}\)

May 6

I finished reading your term papers and your journal entries, and I posted final grades. Have a great summer!

April 29

I posted the homework/quiz average in eCampus.

April 24

We discussed the notion of the fundamental group, and we said good-bye for the semester, since the term paper takes the place of a final exam.

April 22

We classified the letters of the Greek alphabet according to homeomorphism classes and also according to homotopy equivalence classes.
The assignment is to continue working on your paper, which is due on April 25.

April 17

We continued working on the exercise from last time.
The assignment is to continue working on your paper, which is due on April 25.

April 15

We worked in groups on an exercise on the unit circle.

April 10

I posted solutions to the second exam and returned the graded exams. In class, we worked some exercises on homotopy from Chapter 11.
To access the link to turnitin to submit the first draft of the paper, log in to the course in eCampus and click on the “Information” link. There you should see an item labeled “First draft of paper” and a link titled “View/Complete” that leads to a submission screen.

April 8

We worked on connectedness: Exercise 3 on page 191, Exercise 3 on page 196, Exercise 5 on pages 201–202, and Exercise 4 on page 207.
There is no assignment to hand in next time, for you are working on your paper.

April 3

Second exam.

March 27

We discussed the notion of connectedness and worked on some exercises.
Reminder: The second exam, covering Chapters 4, 5, and 7, takes place next Thursday, April 3. The assignment for Tuesday is to make a list for each of these chapters of two important theorems, three important examples, and four important concepts.

March 25

We discussed the preservation of compactness by continuous functions, Tychonoff's product theorem, and the axiom of choice.
The assignment for next time is to write up a solution to Exercise 1 on page 161, which we discussed in class.

March 20

In class, we found that \(\R\) with the standard topology is not compact, but is second countable and hence is first countable and separable and Lindelöf; and \(\R\) with the discrete topology is first countable, is not Lindelöf, hence not compact, is not separable, hence not second countable; and \(\R\) with the half-open interval topology is not compact, is Lindelöf, is separable, is first countable, but is not second countable. The natural numbers with the cofinite topology form a topological space that is compact, hence Lindelöf, and is second countable, hence first countable and separable. You can confirm these statements at Austin Mohr's Spacebook website by searching for “Euclidean Topology”, “Uncountable Discrete Topology”, “Right Half-Open Interval Topology”, and “Finite Complement Topology on a Countable Space”.
The assignment for next time is to continue reading Chapter 7 and to solve Exercise 6 on pages 155–156.

March 18

We discussed the notions of open covers, subcovers, and refinements, as well as compact topological spaces and Lindelöf spaces.
The assignment for next time is to start reading Chapter 7 and to write up a solution to Exercise 2 on page 144 in Section 7.1 (which we worked on in class).
Also, you should be getting started on the term paper. To focus your attention, I would like you to send me an email before April 1 saying what topic you have chosen to write about. And I would like to see a first draft of the paper by April 11, so that I can give you some feedback before the final paper is due.

March 6

The main topic today was the Tietze extension theorem and its proof.
The assignment for Spring Break is to travel safely and to return refreshed.

March 4

We worked on understanding how the separation axioms interact with the subspace topology, the product topology, and the quotient topology. Summary: All of the separation properties can be lost by passing to quotient spaces. Properties \(T_0\), \(T_1\), \(T_2\), \(T_3\), and regularity are inherited by subspaces and are preserved by taking products. Property \(T_4\) and normality are exceptional, for these two properties are not always inherited by subspaces and are not always preserved by taking products.
The assignment is to think about what topic you might like to write about in your term paper.

February 27

We discussed the definitions of the separation axioms.
The assignment for next time is to skim Chapter 5 (in particular, learn the definitions) and to write up a solution to the problem that we worked on during class. Namely, on a set with three elements, there are nine essentially different topologies (which you can find listed in wikipedia). Which of these topologies are \(T_0\)? \(T_1\)? \(T_2\)? \(T_3\)? \(T_4\)? regular? normal?

February 25

We discussed the quotient topology and the concept of homeomorphism.
The assignment for next time is to read Section 4.4 and to do Exercise 2 on page 78 (which we worked on in groups during class). Notice that in part (f), the closed intervals need to be bounded intervals. Indeed, we shall see later that no two of the closed sets \([0,1]\), \([0,\infty)\), and \(\R\) are homeomorphic to each other.

February 20

We discussed the product topology. In particular, we found that when \(\R\) has the discrete topology, the product topology on \(\R\times \R\) equals the discrete topology on \(\R^2\); but when \(\R\) has the cofinite topology (in which closed sets are finite, and open sets are complements of finite sets), the product topology on \(\R\times \R\) is strictly finer than the cofinite topology on \(\R^2\).
The assignment for next time is to read Sections 4.5 and 4.6 and to do Exercise 2 on page 89 in Section 4.6.

February 18

I returned the graded examinations, and we discussed the notion of the subspace topology (or relative topology).
The assignment for next time is to read Sections 4.1 and 4.2 and to write up a solution to the following exercise that we worked on during class: Let \(X\) be the set \(\mathbb{R}\) of real numbers equipped with the standard topology, let \(Y\) be the half-line \([0,\infty)\) equipped with the subspace topology, and let \(A\) be the interval \([0,1)\). Compare the sets \(A^\circ\) (the interior of \(A\)), \(\Cl A\) (the closure of \(A\)), \(\Fr A\) (the frontier of \(A\)), and \(A'\) (the derived set of \(A\)) computed relative to \(X\) with the same sets computed relative to \(Y\).

February 13

The first examination was given. Solutions are available.

February 11

We reviewed for the exam to be given next time on Chapters 2 and 3.
The interesting subtlety mooted near the end of class is that property (i) in the definition of a metric on page 16 can actually be deduced from the other three properties. Namely, \[ 0=D(x,x) \le D(x,y) + D(y,x) = 2D(x,y), \] where the first equality is property (iii), the inequality is the triangle inequality, and the second equality is the symmetry property. Dividing by 2 shows that \(D(x,y)\ge 0\), which is property (i).

February 6

I posted a summary of your discoveries from last class. In class today, we discussed some of these items.
The assignment for next time is to read the first three pages of Section 4.3 (pages 70–72) and to do Exercise 7 on page 75. Of course you should also be reviewing Chapters 2 and 3 in preparation for the examination to be given on Thursday, February 13.

February 4

I posted a solution to Exercise 1 in Section 3.1.
In class, we discussed the notions of interior, closure, frontier (boundary), exterior, and derived set. We worked in groups on determining how these notions interact with the set operations of intersection, union, and complement.
The assignment for next time is to read Section 3.6 and to do Exercise 4 on page 63 in Section 3.6.

January 30

Since I forgot to collect the assignment due today, I will collect the papers next time.
In class today, we worked in groups on an exercise on topologies on a set with three elements. (There are 29 possible topologies all together; 25 topologies for which some point is open; 25 topologies for which some point is closed; and 19 topologies for which no proper subset is simultaneously open and closed.) Arranging the different topologies according to fineness and coarseness requires some sort of tree diagram or branching graph, since there are some topologies that are not comparable to each other (that is, neither topology is coarser than the other).
Incidentally, the problem of enumerating the topologies on a large finite set is difficult. For some additional information about this problem, see sequence A000798 in the On-Line Encyclopedia of Integer Sequences.
The assignment for next time is to read Section 3.5 on derived sets and to do Exercise 3 on page 52 in Section 3.3 and Exercise 1 on page 58 in Section 3.5.

January 29

A good source for additional examples and supplementary reading is the book Topology without Tears by Sidney A. Morris. The book is freely available online.

January 28

I returned the graded papers from the assignment turned in last time.
In class, we discussed the notions of basis and subbasis, and we worked in groups on Exercise 6 from Section 3.1 on page 43 (groups with odd numbers) and Exercise 4 from Section 3.2 on page 48 (groups with even numbers).
The assignment for next time is for each student individually to write up a solution to the problem that your group worked on. Also, read Sections 3.3 and 3.4 (pages 49–54).
Remarks: The topology in Exercise 6 is called the cofinite topology, since the open sets are the ones that have finite complements. The topology in part (a) of Exercise 4 is called the half-open interval topology; one of the amusing properties of this topology is that each set in the indicated basis is simultaneously open and closed with respect to the topology.

January 23

We discussed the notions of convergence and continuity in metric spaces.
The assignment for next time is to read the first two sections of Chapter 3 (pages 40–47) and to solve Exercise 1 on page 43 (in Section 3.1) and Exercise 6 on page 48 (in Section 3.2).

January 21

I posted updates to the website with some instructions about the journals and the term paper. In class, we worked a group quiz and discussed solutions.
The (individual) assignment to hand in next time is Exercise 2 on page 18 in Section 2.1.

January 16

Groups 1–4 presented the concepts of metric spaces, neighborhoods, open sets, and closed sets. We encountered the principle that when you learn a new definition, you should also learn some examples and some non-examples of the concept.

January 15

I spoke on the telephone to the textbook manager at the campus bookstore. He told me that he has ordered an additional 21 copies of the textbook.

January 14

At the first class meeting, we did introductions and then split into seven groups to work on classifying the letters of the Greek alphabet into topological equivalence classes.
Each group has the assignment of becoming expert on the corresponding section of Chapter 2 and solving one exercise to present to the class. Due to the lack of availability of the textbook in the campus bookstore, only a few members of the class have copies at the moment, but one person in each group has a copy of the book.