Last updated Mon Oct 14 18:36 EDT 2019
"Virtual" due-date: Friday, 10/18/19. No homework will be collected that day for this assignment. (It is still possible that I'll have you hand-in some problems for this assignment at a later date, but no earlier than Wed. 10/23/19.)
You are required to do all of the problems below. You will not be required to hand them all in. I will announce later which ones you do have to hand in on the due-date. Don't make the mistake of thinking that I'm collecting only the problems I think are important.
The "due date" above is the date that your written-up problems should be handed in, but don't wait to get started on the assignment. You should always get started on problems as soon as we cover the relevant material in class.
- C:
- Finish reading all parts of Rosenlicht Chapter II of Rosenlicht that we did not cover in class. (This would have been part of Assignment 2, but I didn't want to lengthen that assignment.)
- Read the handout "Interiors, Closures, and Boundaries" (henceforth "the ICB handout") posted on the Miscellaneous Handouts page. (You may ignore fact #13 until we've defined convergent sequences.) Prove all the facts, other than #13, that have not been proven in class as of Wed. 10/16/19. The facts not proven in class will include numbers 4, 10, half of 12, and several from the 14–22 group.
- A: Rosenlicht Chap. III/ 1c, 3, 4, 6, 7.
Note:
- From Chapter III on, you may notice that there is very little correlation between the difficulty of a problem and where it appears in the chapter's list of problems. Some of the earliest problems are among the hardest. The problem-numbering is governed by the order in which topics appear in the chapter, not by difficulty.
- In problems like 4 and 6, keep in mind that "proof by picture" is not a valid method of proof. In these two problems, you will need to show algebraically that open balls of certain centers and radii (which you have to figure out) are contained in certain sets. In these problems you will be tempted to use the concept of "the point (or a point) on a graph that's closest to a given point not on the graph." But you can't assume there is a closest point, unless you have a written proof that such a point exists. (For #6, it's highly unlikely that any attempts you make to prove the existence of such a point will succeed; you don't have the tools yet. Once you do have the tools, later in this course, you'll see that it's circular reasoning to try to use the closest-point idea to prove that the set in this problem is open.) You are likely to find #6 difficult.
By the way, if you're under the impression that the non-book problems are harder than all the book problems, you probably have not been doing all the book problems you've been assigned (or at least haven't been doing them correctly!). For example, you should not have found any of the non-book problems so far to be as difficult as II/10c or III/6. Many of my non-book problems are constructed specifically to give you a better chance of solving some of Rosenlicht's problems.
- In Rosenlicht, \(E^n\) means Euclidean \(n\)-space: the metric space \( ({\bf R}^n,d)\) where \(d=d_{\rm Euc}\) is the Euclidean metric (the one induced by the \(\ell^2\)-norm). (See the last paragraph on p. 34.) So in problems 4 and 6, \(E^2\) is the usual \(xy\)-plane (or \(x_1x_2\)-plane) with the distance-formula that you're used to. In these problems, you may use the notation \((x,y)\) instead of \((x_1,x_2)\), but state that you're doing this (if I have you hand in one or both of these problems), so that I know what you mean from the start.
- B: Click here for non-book problems. Problem B1 is an expanded version of Rosenlicht's problem 1b. For the parts of problems B8–B10 that involve boundaries, you may wait until after the Wednesday 10/16/19 lecture, or use the ICB handout for the facts about boundaries that you need.
In class I gave a hint for B6. The same hint is helpful in B8. Also helpful are a couple of facts on the ICB handout.
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