Math 436
Spring 2018 Journal

Friday, May 4
The final examination was given, and solutions are available.
Tuesday, May 1
On this redefined day, we followed up on the exercise from last time and then played with quotient spaces obtained by identifying edges of rectangles. The slides from class are available.
Monday, April 30
We worked on an exercise about quotient spaces shown on the last page of the slides from class.
Friday, April 27
We discussed quotient spaces (identification spaces). The last page of the slides from class has a review exercise (not to hand in).
Wednesday, April 25
The first four groups presented solutions to the problems from last time. There is no assignment to hand in next class.
Monday, April 23
We worked in groups on six problems, the first three of which fill in the details of the proof from last time, and the other three of which are important exercises from section 7.2. The groups will present their solutions next class.
Friday, April 20
We discussed (without finishing the details) the theorem that a metric space is compact if and only if it is sequentially compact. The last page of the slides from class has some review exercises (not to hand in).
Wednesday, April 18
We discussed relationships between compact sets and closed sets as well as the Heine–Borel theorem for the real numbers. As shown on the last page of the slides from class, the assignment is to write a solution to number 1 in Exercises 7.2.
Monday, April 16
We discussed compactness and the finite-intersection property. As shown on the last page of the slides from class, the assignment is to write solutions to numbers 4 and 6 in Exercises 7.1.
Friday, April 13
We discussed the concept of compactness. The assignment due next class is the exercise on the last page of the slides from class. Of course your solution should include some sentences of explanation.
Wednesday, April 11
The second exam was given, and solutions are available.
Monday, April 9
We reviewed for the exam to be given next class. The slides from class are available.
Friday, April 6
We talked about the order topology and then worked on a review exercise about connectedness and path-connectedness (see the slides from class). The assignment is to study for the upcoming exam.
Wednesday, April 4
By way of reviewing for the exam to be given next Wednesday, we worked in groups on an exercise about continuous functions between topological spaces (see the slides from class). The assignment is to study for the exam.
Monday, April 2
We saw that sequences determine the topology of a metric space. As shown in the slides from class, the assignment due next class is to write a solution to one of problems 1, 2, 3, 4, 6, 8 in Exercises 6.2.
Wednesday, March 28
We discussed a metric on the space of continuous real-valued functions, and we looked at some concrete examples of isometries. As shown on the last page of the slides from class, the assignment is to read section 6.2 in the textbook.
Monday, March 26
We looked at problems 3, 4, 8, and 11 in Exercises 6.1. The assignment due next class is shown in the slide from class.
Friday, March 23
We worked in groups on problems 3, 4, 5, 8, 9, 10, and 11 in Exercises 6.1. The assignment is to write a solution to one of these problems and to read the rest of section 6.1.
Wednesday, March 21
We looked at some examples of metric spaces and discussed the notion of a normed vector space. The assignment due next class is shown on the last page of the slides from class.
Monday, March 19
We discussed the definition of metric spaces and some examples. The assignment due next class is shown on the last page of the slides from class.
Friday, March 9
We discussed the notions of locally connected and locally path-connected. The slides from class are available.
Wednesday, March 7
We discussed the concepts of components and path components and then started working on the assignment due next class, which is shown on the last page of the slides from class.
Monday, March 5
We worked on some exercises from section 5.2. As shown on the last page of the slides from class, the assignment due next class is to write a solution to one of the five problems 3, 4, 5, 6, 9 in Exercises 5.2.
Friday, March 2
The main topic was that continuous functions preserve connectedness. As shown on the last page of the slides from class, the assignment due next class is to read section 5.2 in the textbook.
Wednesday, February 28
We worked in groups on Exercises 5.1. As shown on the last page of the slides from class, the assignment due next class is to write a solution to number 2 or 3 in Exercises 5.1.
Monday, February 26
We discussed the concept of continuity of functions between general topological spaces. As shown on the last page of the slides from class, the assignment due next class is to read section 5.1 in the textbook and to write a solution to number 1 in Exercises 5.1.
Friday, February 23
The first exam was given, and solutions are available.
Wednesday, February 21
We reviewed for the exam on Chapters 1–4 to be given on Friday.
Monday, February 19
We worked in groups on some review exercises shown in the slides from class. I posted some old exams.
Friday, February 16
We discussed the concept of local homeomorphism and worked in groups on number 17 in Exercises 4.1. As shown on the last page of the slides from class, the assignment due next class is to write a solution to parts (i), (ii), and (iii) of number 17 in Exercises 4.1 and to read section 4.3 in the textbook.
Wednesday, February 14
We discussed separation properties and worked in groups on number 13 in Exercises 4.1. The assignment due next class is shown on the last page of the slides from class.
Monday, February 12
We discussed the concept of homeomorphism and revisited the subspace topology. The assignment due next class is shown on the last page of the slides from class.
Friday, February 9
We discussed the subspace topology (relative topology). As shown on the last page of the slides from class, the assignment due next class is to write solutions to numbers 6 and 11 in Exercises 4.1 and to read section 4.2 in the textbook.
Wednesday, February 7
We discussed the concept of connectedness and worked on examples. As shown on the last page of the slides from class, the assignment due next class is to write solutions to number 8 in Exercises 3.2 (about the interaction of interior and closure with union and intersection) and number 6 in Exercises 3.3 (about connectedness and the countable-closed topology) and to read section 4.1 in the textbook (about the subspace topology).
Monday, February 5
We discussed the concepts of neighborhoods, interior of a set, separability of a topological space, second countability, and the Sorgenfrey line. The assignment due next class is shown on the last page of the slides from class.
Friday, February 2
We worked in groups on the six problems in Exercises 3.1 (about limit points). As shown on the last page of the slides from class, the assignment due next class is to read section 3.2 in the textbook about the concept of neighborhood.
Wednesday, January 31
We discussed the concept of subbasis and worked in groups on problems 6–12 in Exercises 2.3. As shown on the last page of the slides from class, the assignment due next class is to write solutions to number 5 in Exercises 1.2 and number 3 in Exercises 2.3 and to read section 3.1 in the textbook.
Monday, January 29
We discussed the concepts of a basis for a topology and the product topology. The assignment due next class is shown on the last page of the slides from class.
Friday, January 26
We looked at some examples in the Euclidean topology on the real numbers, and we discussed the intersection topology (number 7 in Exercises 1.3). As shown on the last page of the slides from class, the assignment due next class is to read section 2.2 in the textbook (about the concept of a basis for a topology) and to write a solution to number 7 in Exercises 2.1 (about closed sets in the Euclidean topology).
Wednesday, January 24
We worked in groups on some problems from Exercises 1.3 about \(T_0\) and \(T_1\) spaces, Sierpiński space, the countable-closed topology, the intersection of topologies, door spaces, and saturated sets. As shown on the last page of the slides from class, the assignment due next class is to read section 2.1 in the textbook (about the Euclidean topology) and to write a group solution to your problem (one paper per group). If you missed class (and so do not have a group), then you may write an individual solution to number 7 in Exercises 1.3.
Monday, January 22
We solved together number 6 in Exercises 1.1 and numbers 2, 3, and 4 in Exercises 1.2. As shown on the last page of the slides from class, the assignment due next class is to read section 1.3 and to write a solution to number 1 in Exercises 1.3.
Friday, January 19
We reviewed from section 1.1 the notions of a topology, the discrete topology, and the indiscrete topology; and we worked in groups on problem 9 in Exercises 1.1. As shown on the last page of the slides from class, the assignment due next class is to write a solution to this problem and to read section 1.2 in the textbook.
Wednesday, January 17, 2018
As a warm-up exercise, we worked in groups on determining the equivalence classes of the capital Greek letters under continuous deformation. As shown on the last page of the slides from class, the assignment due next class is to read section 1.1 in the textbook and to write (or type) solutions to numbers 1 and 4 in Exercises 1.1.
Thursday, January 4, 2018
This site went live today. Once the Spring 2018 semester begins, there will be regular updates about assignments and the highlights of each class meeting.