# Real Analysis vs. Topology

## Main Question or Discussion Point

Hi,

I am currently a sophomore and a math major with thinking of adding computer science as either minor or second major. I get to register for my classes for Fall Quarter in a week, and I am thinking of taking 2 math classes: One will be numerical analysis, and the other is not yet determined. But I am thinking of taking either real analysis or topology, and here are my thoughts for these courses.

-I actually took the first term of analysis last term, but I didn't do very well (ended up taking it Pass/No Pass, and I passed. I also discussed about this in here: https://www.physicsforums.com/showthread.php?t=298030 ). I think that poor performance was partly because of my maturity (whether mathematical or not), and I would really like to take it again someday. For next year, this course is going to be taught by the professor that I had for multivariable calculus, and I liked him quite a bit. I also still own Rudin's PMA as well as Ross's Elementary Analysis: Theory of Calculus, so I might actually consider studying for this course (with my friends who also took analysis last year but dropped after the first quarter), so taking analysis next year might be feasible.

-On the other hand, topology sounds interesting to me too (it actually sounds more interesting than analysis, but I'll never know). I don't know much about the professor who's teaching this course, but he seems to be getting good reviews from his students. But I've heard that topology should be taken AFTER I have taken either analysis or algebra (or possibly both), and I've also heard it's more challenging than analysis or algebra. I don't know how true that is, and I've also heard an argument that says it's not really true.

-It might be normal for me to have algebra as one of my choices, but this class conflicts with my schedule a little bit (I have to sacrifice numerical analysis), and there are some negative feedbacks for this professor (e.g. too difficult, not helpful in office hours, etc), although these feedbacks aren't necessarily from the abstract algebra class (mainly from linear algebra and calculus students). I'm certainly interested in learning algebra as well, but I'm thinking of waiting until my senior year (or super senior year).

Any comment is appreciated, and I'm welcome to answer any question that you might have.

Thanks,

PieceOfPi

I just got done with an undergrad course in topology. I took analysis a couple years earlier. As it turned out, my topology course was pretty easy, using a relatively easy textbook, whereas my analysis course was hard as $%#$ and we used the challenging Rudin PMA. Seeing as how all of chapter 2 in PMA is devoted to topology, I think it actually makes sense to take topology before taking analysis. It seems like this makes for a more gradual increase in difficulty. Seeing as how you already took analysis, I'm not sure if this really is meaningful for you.

Topology seems quite a bit more abstract than analysis, for what it's worth. In topology you talk about very general things like sets and connectedness and compactness whereas in analysis you tend to specialize those results towards things like sequences. Hence topology might be a better preparatory course for another abstract course like abstract algebra.

After taking a bunch of undergrad math I have kind of learned that topology is supposedly crucial for defining the notions of convergence in the topological space. I say "supposedly" because we haven't talked about this very much directly, but I have seen this idea referenced in some more advanced texts. It makes sense because I suppose you need to know what are limit points, what are open sets, all that sort of thing in order to talk about convergence. However I'm still a little bit mystified by this connection.

jbunniii
Homework Helper
Gold Member
Hi,

I am currently a sophomore and a math major with thinking of adding computer science as either minor or second major. I get to register for my classes for Fall Quarter in a week, and I am thinking of taking 2 math classes: One will be numerical analysis, and the other is not yet determined. But I am thinking of taking either real analysis or topology, and here are my thoughts for these courses.

-I actually took the first term of analysis last term, but I didn't do very well (ended up taking it Pass/No Pass, and I passed. I also discussed about this in here: https://www.physicsforums.com/showthread.php?t=298030 ). I think that poor performance was partly because of my maturity (whether mathematical or not), and I would really like to take it again someday. For next year, this course is going to be taught by the professor that I had for multivariable calculus, and I liked him quite a bit. I also still own Rudin's PMA as well as Ross's Elementary Analysis: Theory of Calculus, so I might actually consider studying for this course (with my friends who also took analysis last year but dropped after the first quarter), so taking analysis next year might be feasible.

-On the other hand, topology sounds interesting to me too (it actually sounds more interesting than analysis, but I'll never know). I don't know much about the professor who's teaching this course, but he seems to be getting good reviews from his students. But I've heard that topology should be taken AFTER I have taken either analysis or algebra (or possibly both), and I've also heard it's more challenging than analysis or algebra. I don't know how true that is, and I've also heard an argument that says it's not really true.

-It might be normal for me to have algebra as one of my choices, but this class conflicts with my schedule a little bit (I have to sacrifice numerical analysis), and there are some negative feedbacks for this professor (e.g. too difficult, not helpful in office hours, etc), although these feedbacks aren't necessarily from the abstract algebra class (mainly from linear algebra and calculus students). I'm certainly interested in learning algebra as well, but I'm thinking of waiting until my senior year (or super senior year).

Any comment is appreciated, and I'm welcome to answer any question that you might have.

Thanks,

PieceOfPi
What do you think was the main reason for your difficulty with analysis? If you feel you need more experience with proofs in general (both understanding and producing them), then trying to tackle analysis again at this point would be premature.

Either topology or linear algebra would be a better choice in that case. That assumes it's a point-set topology course and that your experience with analysis gave you some intuition/motivation at least for metric spaces. Even in that case, a rigorous proof-oriented course in linear algebra might be an even better place to start.

Meanwhile, as far as analysis is concerned, Rudin is a pretty tough book if it's your first exposure to analysis and/or rigorous calculus. It's not a great place to start, in my opinion, if you aren't already pretty comfortable with epsilon-delta arguments and filling in the unstated details in slick, terse proofs.

If that is your situation, I highly recommend spending some time reading and doing exercises from either a rigorous calculus book like Spivak or an easier elementary analysis book like Bartle or Pugh. Once you're comfortable at that level, Rudin will be a lot more approachable, and instead of cursing his austere exposition, you may well find it incredibly clear and beautiful. (I know I did!)

mordechai9: Which textbook did you use? It's likely that our topology class is going to use Munkre's Topology, but do you think this textbook is similar in terms of difficulty as yours?

jbunniii:

What do you think was the main reason for your difficulty with analysis? If you feel you need more experience with proofs in general (both understanding and producing them), then trying to tackle analysis again at this point would be premature.
I think the main difficulty was that I had a hard time understanding every definition and theorem in the book. I just felt there were too many, and some of them weren't very intuitive.

Either topology or linear algebra would be a better choice in that case. That assumes it's a point-set topology course and that your experience with analysis gave you some intuition/motivation at least for metric spaces. Even in that case, a rigorous proof-oriented course in linear algebra might be an even better place to start.
I believe the topology course I'm considering of taking is a point-set topology (or at least starts with a point-set topology). I've taken the elementary linear algebra, but I'm actually considering taking advanced linear algebra at some point as well (text: Axler: Linear Algebra Done Right).

Meanwhile, as far as analysis is concerned, Rudin is a pretty tough book if it's your first exposure to analysis and/or rigorous calculus. It's not a great place to start, in my opinion, if you aren't already pretty comfortable with epsilon-delta arguments and filling in the unstated details in slick, terse proofs.

If that is your situation, I highly recommend spending some time reading and doing exercises from either a rigorous calculus book like Spivak or an easier elementary analysis book like Bartle or Pugh. Once you're comfortable at that level, Rudin will be a lot more approachable, and instead of cursing his austere exposition, you may well find it incredibly clear and beautiful. (I know I did!)
I actually took a course called "Elementary Analysis" with the textbook of Ross (the one I mentioned above). Although we didn't do very chapters from this textbook (e.g. Differentiation and Integration), so maybe I should read this textbook from the cover to cover on my own? Or should I read another textbook like the ones you've mentioned?

Once again, I appreciate your comments a lot, and I'm certainly open for more.

jbunniii
Homework Helper
Gold Member
Munkres is a great book for point-set topology, very much the standard choice and deservedly so. It's very well written and extremely clear, not as slick or terse as, say, Rudin's analysis book. I wouldn't call it an "easy" book, but it's very user-friendly, if that makes sense.

I think the main difficulty was that I had a hard time understanding every definition and theorem in the book. I just felt there were too many, and some of them weren't very intuitive.
Did you take the course that used Ross before or after the one that used Rudin?

I'm not familiar with Ross, but I just looked at the table of contents on Amazon and it looks like it covers a lot of the same stuff as Rudin, provided that you read the whole thing including the starred sections. The exercises certainly look easier than Rudin's, and I see that it also gives hints and answers for some/all of them.

I think it would be a great idea to read that book cover to cover and do as many exercises as you can. (As with most mathematics, you won't really learn it unless you do lots of exercises.) If you do that, Rudin still won't be a walk in the park, but you should be very well prepared for the challenge.

I've taken the elementary linear algebra, but I'm actually considering taking advanced linear algebra at some point as well (text: Axler: Linear Algebra Done Right).
Outstanding choice in my opinion. That's a great book. Axler is very easy to read and his proofs are very clean; he makes it all seem very easy. He even adds "(as you should verify)" to point out when he is intentionally leaving a gap in a proof for you to fill in. Come to think of it, just about every line of Rudin should be treated as if it ended with "(as you should verify)".

But just reading Axler's exposition can deceive you into thinking that you understand it better than you do. The exercises aren't too hard (though you'll probably bang your head against a few of them for a while before the ah-ha moment hits) but they'll ensure that you understand the concepts. Try to do them all if you have the time.

I actually took a course called "Elementary Analysis" with the textbook of Ross (the one I mentioned above). Although we didn't do very chapters from this textbook (e.g. Differentiation and Integration), so maybe I should read this textbook from the cover to cover on my own? Or should I read another textbook like the ones you've mentioned?
I think Ross looks like a good place to start, especially since you already own it.

What exactly is meant by Real Analysis? Do you know what book will be used there? Since under Real Analysis, I would normally imagine a course going along the lines of Rudin's Real & Complex Analysis book - i.e. Measure & integration theory, some basic functional analysis etc., which is quite difficult if you don't have much experience with proofs (actually without a course on analysis like Baby Rudin, I'd say it's next to impossible). On the other hand some people would call a course on calculus "real analysis", so check what will be really covered.

Topology following Munkres is pretty basic (at least compared to Real analysis in my understanding), meaning that there are not many prerequisites, although it is good to know some basic calculus to know why you're dealing with some objects, but it's not a neccessity.

It also depends on what courses you want to take afterwards. Topology is used in basically all parts of mathematics, but I think you can learn it also on your own, when you'll need it, since it is not that demanding. But if you're considering going into Differential geometry or Algebraic topology, it's better to take a course on Topology first.
On the other hand, analysis is much harder to learn on your own and it's also a part of general knowledge of a mathematician, so I myself would probably lean more to that course (but remember what I wrote above).

What exactly is meant by Real Analysis? Do you know what book will be used there? Since under Real Analysis, I would normally imagine a course going along the lines of Rudin's Real & Complex Analysis book - i.e. Measure & integration theory, some basic functional analysis etc., which is quite difficult if you don't have much experience with proofs (actually without a course on analysis like Baby Rudin, I'd say it's next to impossible). On the other hand some people would call a course on calculus "real analysis", so check what will be really covered.
I'm talking about analysis with Baby Rudin.

Topology following Munkres is pretty basic (at least compared to Real analysis in my understanding), meaning that there are not many prerequisites, although it is good to know some basic calculus to know why you're dealing with some objects, but it's not a neccessity.
Do you think Ross's Elementary Analysis: The Theory of Calculus is a good place to review some basic calculus that you've mentioned? This book has some stuff on metric spaces as well.

It also depends on what courses you want to take afterwards. Topology is used in basically all parts of mathematics, but I think you can learn it also on your own, when you'll need it, since it is not that demanding. But if you're considering going into Differential geometry or Algebraic topology, it's better to take a course on Topology first.
On the other hand, analysis is much harder to learn on your own and it's also a part of general knowledge of a mathematician, so I myself would probably lean more to that course (but remember what I wrote above).
Basically, I'm still looking into three fields (and possibly going into grad school in one of these): Pure math, applied math, and computer science. So I'm planning to "sample" each class next year, and that's why I'm thinking of taking either topology or analysis (pure math), numerical analysis (applied math), and one computer science course. I should probably have a clear idea of what I really want to do by next year.

I would then go for the Real Analysis course. Having a solid background in analysis is important for any mathematician, topology can be learned along the way if you'll need it. In a course going in the lines of Rudin's PMA, you would actually learn also quite a deal of topology (metric spaces, compactness, connectedness, ...) and at the same time you'll see directly what are all these topological things good for.

I'd go with analysis. Try reading the first two or three chapters of Rudin over the summer just to get the basics down (make sure you do the exercises). Pay close attention to chapter 2 on basic topology. Use Ross as a companion book, i.e. if you can't understand something in Rudin, look it up in Ross and see if that makes it any easier. It won't be easy, but there's really nothing to be gained from postponing analysis. Indeed, I'd say that the topology you learn from Rudin will make your topology class easier when you end up taking it.

It's good to see the opinion from the both sides. I hear people say "analysis gives a good idea about topology" as well as "topology gives a good idea about analysis," so it's hard for me to know which should come first. I do remember, however, there was quite a bit of basic topology involved in analysis, and I think that gave me quite a bit of trouble in that class. So partly I feel like it might be a good idea to understand topology before analysis, although I have a feeling the topology I see in the actual topology class can be A LOT harder than the one I saw in Baby Rudin.

So based on what I've read here, my plan is to read Ross and Rudin (chs. 1-3) over the summer since it would probably benefit me even if I'm going to take topology, and might be a good practice for reading a math textbook or writing proofs. I'd also like to get better writing proofs so... is there a book for that besides Rudin or Ross?

As of now, I'm bit more inclined toward topology (Coffee can be a doughnut, eh?), but I also miss the professor who's teaching analysis next year. It's a bit of a dilemma, I gotta say, but at least I still got 2 more years of undergraduate (3 if I decided to become a super senior), so hopefully I can pick whatever is the best for me.

Thanks for your comments! And I'm open for more if there is any.

-PieceOfPi

Tao's Analysis starts from the very foundations and in doing so, forces you to develop proof skills in a very intuitive environment (natural numbers, integers, etc.). I've fallen in love with Pugh's book. And Munkres is great. In fact, it's a better first book than baby Rudin since it does basic stuff properly (that long chapter 1). It doesn't really matter which you do first.

But really, Rudin's PMA isn't as tough as it's made out to be. True, it can be quite shocking as a first book. But once you get used to it, you'll wish every book were that condensed. And then you read grad books and realise that Rudin's PMA isn't that terse.

If you decide on real analysis then take a look at "Real Mathematical Analysis" by Pugh. Far better than Rudin imo.

I hear people say "analysis gives a good idea about topology" as well as "topology gives a good idea about analysis," so it's hard for me to know which should come first.
-PieceOfPi
I would say this more precisely: you certainly don't need analysis for topology, but in order to understand why you're dealing with some subjects, it's good to know some basic applications, which at the most elementary level are usually found in analysis. On the other hand in topology you will learn much more than you will actually need in your analysis class and mostly things that are actually used as late as in an algebraic topology/differential geometry class, which are usually taught as graduate courses.

I personally wouldn't postpone analysis, because for an undergarduate it's a far more important class than topology. At my university, math majors usually take a Rudin analysis course in their first or second year and topology in their third year, usually just before taking a class on differential geometry or simmilar, and I also find this as the natural order.