Posted May, 2. We will start the meeting on Thursday, May 3 with the Perfect Set Theorem #64, and then do the remaining problems #65 - #70 in the notes in the remaining time and in our last meeting on Tuesday, May 8.
Posted April, 27. In the meeting on Tuesday, May 1, we will do as much as we can of the following:
- Go over and make sure we all understand Problems #57 - #59 on continuous functions on Baire space.
- Do #60, the combinatorial characterization of compactness.
- #61 and #62, which is Theorem 10.19 in the book. This is the main idea used in the proof of the Perfect Set Theorem, so it is important that it is understood.
- #63 and #64, the Perfect Set Theorem.
Posted April 25. In the meeting on Thursday, April 26, I would like to do the following, in order:
- Review the proof of #48, the uniqueness of the Cantor-Bendixson decomposition. Please claim this if you can do it and have not claimed it yet, it would be good to have more people work through the proof.
- Review the proof of #56, Cantor's proof of the existence of the Cantor-Bendixson decomposition. Please claim this if you can do it and have not claimed it yet, it would be good to have more people work through the proof.
- Do #57 -- #59.
- Do #60 and #61, if it is possible.
Posted April 22. In the first hour of class on Tuesday, April 24, we will review #47 and #48 to make sure that the proof of the Cantor-Bendixson Theorem is understood, and then cover as many of the problems #50 - #52 and #53 - #56 (in the updated Notes) as possible, so please send claims if you do any of these by Tuesday. My aim is to spend the second hour explaining some of the material in 10.14 - 10.19 of the book, but we will see whether this is feasible.
Posted April 18. The notes have been updated with a sequence of problems breaking down the proof of the Cantor-Bendixson Theorem and related statements. Some of them are very easy. Try to work on these problems before the class meeting on Thursday, April 19, at least to get used to the terminology: the plan is to do as much as we can on Thursday and aim to complete this part of the Notes by next Tuesday.
Posted April 14. Through the last class meeting on Thursday, 4/12, we have presented or discussed most of Problems #1 -- #36, and that's all we will do with these. We will skip #37, but will do #38 and discuss #39, so claim these two problems if you can do them no later than Monday, 4/16 evening.
Note. I mentioned on Thursday that Baire space is "homeomorphic" with the set of irrational real numbers and tried to explain how one could go about proving this without explaining what exactly it means---and, in fact, the argument I was trying to outline is neither simple nor entirely correct (as I explained it). So forget about this for now: I'll come back to it at some time, do a better job of it and give references where you can find a complete proof ---it is an important result.
Before Tuesday, 4/16, you should also
- Read carefully through 10.8 in Chapter 10 of NST. It contains the definitions that we need to go on, some of which I talked about in the last lecture.
- Do and claim by Monday evening as many of Problems #40 -- #42 as you can.