We’ll use this page to keep track of what has happened each day in class. It won’t contain any of the nitty-gritty details, but will instead serve to summarize what has transpired each day.

Week 1

  • Monday, January 14: First day! The first few minutes of class were devoted to me attempting to learn names. I think I got them all! Next, we did a very quick tour of BbLearn and the course website. The remainder of the class meeting was devoted to discussing isomorphisms as they occur in multiple mathematical contexts. We concluded with an example of a continuous bijection whose inverse is not continuous.
  • Wednesday, January 16: After answering a few quick questions about the syllabus, we jumped into student presentations. We had AS, FA, JH, WC, and SM present Theorem 2.2, Exercise 2.3, visual for Exercise 2.3, Theorem 2.6, and Theorem 2.7, respectively. We ran out of time for Theorem 2.8 and we will address that one at the beginning of class on Friday. Great first day of student presentations!
  • Friday, January 18: Another good day of student presentations. We did not get through everything that was planned, but we should be able to catch up. We had RD, PB, JN, and MS present Theorems 2.8, 2.9, 2.10(forward implication), and 2.10(reverse implication), respectively.

Week 2

  • Monday, January 21: MLK Day, no class.
  • Wednesday, January 21: We devoted the first few minutes of class to discussing the relationship between the Axiom of Choice, the Well Ordering Theorem, and Zorn's Lemma, as well as the relationship between the Well Ordering Property and the Axiom of Induction. Next, we had RB/JW, QC, and PH presented Theorems 2.11, 2.12, and 2.13, respectively. I'm really pleased with how things are progressing.
  • Friday, January 18: After discussing the basics of LaTeX, we divided the class up into five small groups, were each group was tasked with putting their proposed proof to an assigned theorem on the board. We had AB, JB, and PB share their group's proof for Theorems 2.14, 2.16, and 2.18, respectively. We didn't have time to discuss Theorems 2.19 and 2.20. We'll kick off with Theorem 2.20 next time.

Week 3

  • Monday, January 28: First, we had MS and VM present Theorems 2.20 and 2.21, respectively. Then we had EP and RB present Theorems 2.19 and 2.22, respectively. This was followed with a quick summary of the Schroeder-Bernstein Theorems. With the time we had left, we discussed the definitions of topology and topological space and tinkered with some examples.
  • Wednesday, January 30: AB, JH, FA, RD, AS, and ZF presented Theorem 3.3, Exercise 3.6(discrete), Exercise 3.6(indiscrete), Exercise 3.6(co-finite), Exercise 3.6(co-countable), and Exercise 3.7, respectively. Along the way, we also discussed Exercises 3.1, 3.2, 3.3, and 3.4.
  • Friday, February 1: We divided the class up into a few small groups and then had RD, SM, PH, EP, RH, and QC present Exercise 3.8(indiscrete and co-finite), Exercise 3.8(standard and discrete), Exercise 3.10, Exercise 3.11(1)(2), Exercise 3.11(3)(4), and Exercise 3.12(1)(2), respectively.

Week 4

  • Monday, February 4: We didn't cover much ground today, but that's okay. We spent a good chunk of time reviewing the key concepts related open and closed sets and then I presented a proof of Theorem 3.13. We took our time with this, which didn't leave a lot of time for much else. We had JW present Theorem 3.14. The second implication in Theorem 3.14 was a little rushed due to time constraints, so we will likely revisit it next time.
  • Wednesday, February 6: The first few minutes were devoted to summarizing some recent key concepts and setting the stage for what is coming later in the semester. We had JW, JN, and YD present Theorem 3.14, Theorem 3.15, and Theorem 3.16, respectively. After the student presentations, we chatted about Exercises 3.17 and 3.18 together as a class.
  • Friday, February 8: Another busy day. We had JB and AS/JW present Theorem 3.20 and part (1) of Theorem 3.22, respectively. Along the way, we discussed Exercises 3.19 and 3.21 together as a class.

Week 5

  • Monday, February 11: Well, we're behind schedule again, but that's okay. We had RH present part (2) of Theorem 3.22, which provided some rich discussion. This was followed by a presentation of Exercise 3.23 by ZF. We quickly discussed Exercise 3.24 as a class and then YD presented a proof of Theorem 3.26. We will pick up with Exercise 3.27 next time.
  • Wednesday, February 13: After discussing bases of topological spaces, we had SM and RB present Exercise 3.27 and Theorem 3.28, respectively. Next, we discussed Exercises 3.29 and 4.2 together as a class.
  • Friday, February 15: Today was all about making sure everyone is up to speed. We had FA present Theorem 3.30 and then we slowly discussed the proof of Theorem 4.1 and most of the proof of Theorem 4.3.

Week 6

  • Monday, February 18: We kicked off by wrapping up the proof of Theorem 4.3 and then we discussed Exercises 4.4, 4.5, and 4.10.
  • Wednesday, February 20: After enjoying "Hitler Learns Topology", we had a quick discussion of Exercise 4.10 (which we will come back to after the exam) and then had PB, VM, and WC present Theorem 4.25, Exercise 4.26, and Exercise 4.27, respectively.
  • Friday, February 22: Snow day!

Week 7

  • Monday, February 25: Students took the in-class portion of Exam 1.
  • Wednesday, February 27: The entire class period was devoted to students working on and discussing the take-home exam.
  • Friday, March 1: We revisited Exercise 4.10 and then discussed Theorem 4.28.

Week 8

  • Monday, March 4: After handing back the exams, we discussed the various separation properties. Next, I divided the class up into several small groups, each tasked with discussing one of the problems due today. We summarized Theorem 5.1 and Exercise 5.2 and then had RD present Exercise 5.5. This was followed by quick presentations of parts (1) and (2) of Exercise 5.6 by QC and RH, respectively.
  • Wednesday, March 6: We had YD, VM, EP/AS present Exercise 5.6(3), Exercise 5.6(4), and Theorem 5.8, respectively. I'm really happy with how things are going.
  • Friday, March 8: After quickly discussing Theorem 5.19, parts (1) and (3) of Exercise 5.12, and Exercise 5.14, we divided the class up into several small groups, each tasked with discussing Exercise 5.13. We came together after several minutes and filled out the table for Exercise 5.13 as a class.

Week 9

  • Monday, March 11: After handing back the take-home portion of Exam 1, we briefly discussed the last question on the exam. Next, we revisited part (2) of Exercise 5.12, which was left over from last time. Then AB presented Theorem 5.16 and RD got us started on Theorem 5.17.
  • Wednesday, March 13: RD wrapped up the proof of Theorem 5.17 and then we finally had JN present the last part of Exercise 4.10. We finished up by discussing Exercise 5.18 together.
  • Friday, March 15: Busy day! We had MS, JB, PH, PB, and WC/JH/ZF/QC/SM/FA present Theorem 5.19, Theorem 5.20, Theorem 5.23, Exercise 6.1, and Exercise 6.2, respectively.

Week 10

  • Monday, March 25: Today we spent some time reviewing and then I lectured over Exercise 6.3, Exercise 6.5, Theorem 6.5, and Theorem 6.9. Along the way, we introduced the notion of second countable.
  • Wednesday, March 27: We split the class up into small groups, where each groups was responsible for discussing the solution/proof to a designated problem. We had PB, MS, WC, PH, and AS present Exercise 6.10(2), Exercise 6.10(3), Theorem 6.11, Exercise 6.12, and Exercise 6.13, respectively. We had a few minutes left at the end, so we started discussing compact spaces.
  • Friday, March 29: While half of us were away at a conference, David Deville covered for me. EP, ZF, PB, and PH presented Theorem 7.1, Theorem 7.2, Theorem 7.3, and Corollary 7.4, respectively.

Week 11

  • Monday, April 1: We started by reviewing the concept of compact by playing with some examples of spaces that are not compact. Next, we revisited the the proofs of Theorem 7.2 and Theorem 7.3. Then we divided the class up into small groups, where each groups was tasked with discussing the proofs of Theorem 7.5 and Theorem 7.6. With the few minutes we had left, I sketched the proof of the forward implication of Theorem 7.6.
  • Wednesday, April 3: JW, JN, AB, and PH presented Theorem 7.6, Exercise 7.7, Theorem 7.8, and Theorem 7.9, respectively.
  • Friday, April 5: After a fun discussion about applications of topology, I presented two different proofs (the first was only the sketch of an argument) of Theorem 7.14.

Week 12

  • Monday, April 8: Busy day! We had JB, FA, YD, and JH present Theorem 7.12 (regular), Theorem 7.12 (normal), Theorem 7.15 (forward implication), and Theorem 7.15 (reverse implication).
  • Wednesday, April 10: To give students a break, I lectured over 7.16, 7.19, and 7.19
  • Friday, April 12: The students took the in-class portion of Exam 2.

Week 13

  • Monday, April 15: While the students are working on their take-home exams, I'll lecture for a few class meetings. Today, we started Chapter 8.
  • Wednesday, April 17: More lecturing. We continued where we left off in Chapter 8.
  • Friday, April 19: We picked up where we left off on Wednesday and continued discussing Chapter 8.

Week 14

  • Monday, April 22: Last day of lecturing for a bit. We got through Theorem 8.26.
  • Wednesday, April 24: We divided the class up into several small groups, each tasked with coming to consensus on one of the problems from the homework. We had RB, JH, AS, RH, PB, QC, and JB present Exercise 8.27, Theorem 8.28 ((a) implies (b)), Theorem 8.28 ((b) implies (c)), Theorem 8.28 ((c) implies (a)), Theorem 8.29, Exercise 5.30, and Corollary 8.31, respectively.
  • Friday, April 26: We split the class up into several small groups and then had SM, RD, AS, FA, and WC present Theorem 8.32(continuous), Theorem 8.32(onto), Theorem 8.32(open), Exercise 8.34, and Theorem 8.35, respectively.

Week 15

  • Monday, April 29: We kicked off with ZF presenting the proof of Theorem 8.36 and then I discussed homeomorphisms.
  • Wednesday, May 1: I lectured over Exercise 8.44, Exercise 8.45, Exercise 8.46, and Theorem 8.47.
  • Friday, May 3: Last day! I lectured over metric spaces.

Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Google Scholar
  Impact Story

Current Courses

  MAT 411: Abstract Algebra
  MAT 690: Genome Combinatorics

About This Site

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Land Acknowledgement

  Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.