Difficulties with Chapter 2 in Rudin

Question

I have been reading Rudin (Principles of Mathematical Analysis) on my own now for around a month or so. While I was able to complete the first chapter without any difficulty, I am having problems trying to get the second chapter right. I have been able to get the definitions and work out some problems, but I am still not sure if I understand the thing and it is certainly not internalized.

I am wondering whether I should take this shaky structure with me to the next chapters, hoping that the application there improves my understanding, or to stop and complete this chapter really well?

What do you think?

As for my background, I am quiet close to completing Linear Algebra by Lang (having done a course in Linear Algebra from Strang). I have completed Spivak's Calculus. I come from an engineering background and so I have done multivariable calculus, fourier analysis, numerical analysis, basic probability and random variables as required for engineering. One of the professors advised that I may be better off studying Part I from Topology and Modern Analysis by GF Simmons, but I am finding that completing that book itself may take a semester and I would prefer not to wait that long to start with analysis.

Thank You

EDIT: If it makes any difference, I am studying on my own.

EDIT: So, I have accepted the answer by Samuel Reid. I too have found the limit point definition as illustrated by Rudin and the large set of definitions listed there somewhat dry and without any motivation or examples. This is one of the places in the book which makes it a little difficult for self-study. What I found working in this case is, taking some drill problems from other books and working through them. I will advise anyone to go real slow over the sections 2.18 to 2.32 . There are too many definitions and new concepts in that sections and to miss even one means you cannot move forward. To tell the truth, I found Simmons's 50 pages (from chapter 2 section 10 to the end of chapter 3) to be more useful than the corresponding 4.5 pages in Rudin.

Answer 1

As I found out while working through that chapter, a lot of misunderstanding can arise from not understanding the idea of a limit point thoroughly. To remedy this, I recommend you visit a question I asked a little while ago: Understanding the idea of a Limit Point (Topology).

Another tip would be to remember that in most situations you do not need to worry (conceptually) about the full definition of compact as the whole "open cover containing a finite subcover" which is a loaded statement as the definition of open cover trickles down back to the limit point. Just remember that compact can sometimes be visualized as "closed and bounded". When I was working through this chapter it helped me to try and draw out pictures for the concepts (and then I would make them up in Adobe Illustrator as you can see in the link above). Once you have some sort of solid mental imagery for a particular concept it will be easier to build on the previous terminology when a new concept is introduced. A first exposure to Topology is extremely difficult in this respect as there is so much new terminology that you are not familiar with, and then they build on it immediately!

A few seemingly unimportant things I would suggest that you should NOT gloss over.

• "Open relative to..." (Theorem 2.30, Theorem 2.33)
• Sequences of sets, intervals, k-cells, etc. (Theorem 2.38, Theorem 2.39)
• Specifically, make sure you understand Theorem 2.41 as it really wraps up a lot of the concepts in this chapter and tests that you actually do understand the notions of closure, boundary, compactness, limit points, and how they relate to each other.

Of specific importance if you plan on studying Convex Geometry (Convex Sets, Convex Polytopes, etc.) it is very important that you have a great understanding of anything "Open relative to..." or "Compact relative to..." as they lead to an understanding of the style used in the basic foundations of convex geometry in relative interior, relative boundary, etc.

---

I would highly recommend that you spend more time going over the material in the actual book and do as many exercises as you can. I found that with this book in particular, you think you understand the meaning of a particular Theorem or think you understand why some result is important, only to be blown away during an exercise when you realize that the theorem means something different than you thought it did. Make sure you that you can get through some (if not most) of the exercises before moving on to the next chapter and if you are struggling with one, CONTINUE TO STRUGGLE WITH IT! Only post on here as a sort of last resort if you have spent maybe 3-4+ hours on a single question and can't make any progress. Remember to hop around on the questions for a bit, if you've solved maybe 50% of them and the remaining questions all seem very hard, try one for 15-20 minutes, go to another one and try it for 15-20 minutes, and keep switching around on the questions (pretend it's like the putnam!); I find that things click faster for me that way. If I'm hopping around between 5 or 6 questions and spend 4 hours working on them I'll likely be able to solve 2 or 3 and if I just ram my head against a wall on one of them for 4 hours I might not even solve that one. Keep in mind that there are certain exercises (a few each chapter) that are VERY hard, so don't get discouraged! Stay passionate about the concepts and don't worry if things aren't obvious... because they aren't. Remember it took some of the greatest geniuses of the past few generations to figure out this stuff in the first place!

Good luck!

Answer 2

This approach may not suit you, but I definitely found that it helped me. My suggestion while studying Rudin chapter two is to look at Munkres' Topology chapters 2 and 3. What I found was though general topological spaces are in a more abstract setting, they it gave me a lot of motivation. For example if you're having trouble on the bit about open sets relative to the metric space embedded in, then you can look for motivation in the subspace topology.

The idea is the following: If you are studying material that is at level zero (metric spaces), then build up enough machinery at say level ten (topological spaces) to tackle all the level zero problems. The advantage of this is sometimes - not always though - when you are solving a problem at level ten you can look for specific cases in level zero and understand why the problem is true/false there. If you are working at a lower level on the other hand, it is more difficult to ascend up to another level because you will have to learn new material.

As an example, suppose you know the definition of a continuous function in terms of open sets. Suppose you want to show that the zero set of a continuous function is closed. Then because the singleton set $\{0\}$ is closed in a metric space its pre-image must be closed too.

The high level approach to this is as follows: If $f,g$ are continuous functions from $X$ to $Y$ that are topological spaces, $Y$ being Hausdorff then the equaliser

$\Delta = \{ x\in X : f(x) = g(x) \}$ is closed in $X$. Once you know how to do this, whatever you wanted to prove above is just a special case by setting $g(x) = 0$ and noting that every metric space is Hausdorff.

Regarding your difficulties with the material, here are some exercises that may help with your understanding. It requires a little more than the material of chapter 2 of Rudin (You just need to know what cauchy sequences and complete metric spaces are). I promise you if you do the below, your understanding of the material will be strengthened.

(1) Suppose that $(X,d)$ is a complete metric space and $I_n$ a sequence of non-empty closed sets such that $I_n \supset I_{n+1}$ for all $n \geq 0$ and diam $I_n \rightarrow 0$ as $n \rightarrow \infty$. Then $$\bigcap_{i=0}^\infty I_n$$ consists of exactly one point. The diameter of a subset $A \subset X$ is

diam $A :=$ sup $\{d(x,y) : x,y \in A\}.$

(2) Using (1) above, prove the following version of the Baire Category theorem: If $\{G_a\}$ is a countable collection of non-empty, dense open sets in a complete metric space $X$, then $$\bigcap_{a} G_a \neq \emptyset.$$

(3) Finally using (2) above, prove the following: Let $\{H_a\}$ be a countable collection of nowhere dense sets in a complete metric space $X$. Prove that there is a point of $X$ that is not in any of the $H_a's$.

You can post your answers here, and I will try and help you out with them.

Have a good day!

Answer 3

I was in precisely your situation several years ago. In hindsight, Rudin was a poor text for self-study. Perhaps if you're the kind of person who grew up with mathematical culture (parents mathematical, friends interested in mathematics,etc), you'll have the broad cultural background necessary to appreciate the overall approach, or a network of people to bounce ideas off of.

For me, working to learn formal mathematics with my background in electrical engineering was an uphill fight. Rather than struggling through Rudin in a linear fashion, I'd recommend supplementing the exercises in Rudin with other texts that provide more examples, and a more complete link to the history and context in which the subject developed.

I found the exposition in Thomas Korner's book, A Companion to Analysis especially helpful. I agree with the other poster that the Munkres book is excellent. Also, for whatever reason, I found the Royden book better for self-study than the Rudin text. There are also numerous expository texts on introductory analysis (the subject is more-or-less standard, modulo pedagogical preferences) that attempt friendly introductions; I won't link to any here since I'm not familiar with any in particular, but a quick google search would surely bring up a few.

Rather than trying to tease understanding out of unchanging paragraphs, I recommend you supplement Rudin with many other texts. Because the subject is relatively standard, other texts may provide a different perspective on a topic that provides you with that "ah-ha" moment -- why have one teacher when you can have many? Think of those Rudin paragraphs as research problems unto themselves, and go hunting for the context you need to place them in perspective.

Finally, I don't think it's at all a bad idea to skip ahead in Rudin to the next chapter, or to whatever part you're interested in. Working on a problem in the next chapter may provide the context you need to understand the need for the topological background in chapter 2.

Good luck!