Topology vs Analysis, which should be studied first?

[ahsanxr]

So I'm planning to delve into both of these subjects in some depth during the summer to prepare for undergrad analysis (using rudin) and a graduate differential topology class. My question is which one should I start out with and pay more attention to. I obviously need to study a lot of topology if I am to comprehend that diff top class but I've heard conflicting views on which should be studied first, for example,

Micromass says:

Personally, I would take analysis before you take topology. Make sure you know about metric spaces before topology (metric spaces are usually studied in analysis, but not always).

whereas Monocles says:

I took my first analysis class this past fall semester, and I actually studied topology first. Analysis is a pre-req for topology at my school, but I had some free time over the summer and really wanted to start with graduate level topology this fall, so I studied undergrad topology on my own from Munkres and consulted one of the professors who had taught it recently to check some of the work I did and answer questions. I also didn't have much math background before this - just 2 proof based classes before this (linear algebra and abstract algebra).

I found it challenging and very fun. It probably helped that I already had a strong interest in topology, in the sense that I had been learning about how manifolds are used in physics before then. I didn't find the lack of motivation from not having taken analysis too much of roadblock - and it did help motivate analysis when I took it this fall! It also made analysis really easy. At my school, analysis is thought to be quite hellish, but the majority of the problem solving techniques you require you'll have picked up from topology, making analysis a walk in the park.

So what are your opinions on this?

As for texts, I am looking at Munkres for Topology and Pugh for Analysis.

 

[mwalmasri]

Hi,
you should first study Real and complex analysis and after that you go to Topology and after that you study Functional analysis

 

[morphism]

You should be aware that in the two passages you've quoted, topology most likely refers to point-set topology, i.e. the subject where the notion of a "topology on a set" is defined, and various basic things such as compactness, connectedness, etc. are studied. Differential topology is a completely different beast. Do you know what this course will cover? Because it might very well be the case that you'll need more than just topology and analysis to understand what's going on.

Anyway, to answer your question: it's definitely a good idea to learn about metric spaces, open and closed balls, and things like that before learning about general topological spaces, i.e. I tend to agree with micromass.

 

[micromass]

Let me motivate my decision a bit.

Usually, the first sentence in a topology course will be:

Definition: A topological space is a set X together with a set of subsets $\mathcal{T}$ such that blablabla is satisfied. The elements of $\mathcal{T}$ are called open sets.

If you have no idea what open sets are supposed to represent, then you're basically screwed at this point. It will take you quite some time before you actually find out the point of topology. The course will go on and define the closure, closed sets, interior, boundary, convergence, etc. You'll have no idea what you're doing if you never met these concepts before.

This is why you should be familiar with metric spaces, since this is the usual context where those concepts are introduced.

However, if you're already familiar with open sets in $\mathbb{R}$, then topology will be possible to study. If you never heard of the term open set before, then you should not study topology.

I think it is actually certainly possible to do topology without reference to metric spaces or analysis. But that approach requires a very very careful motivation of all the concepts in the course. I don't think such a course actually exists at this point. Perhaps it's a book I should write once

So, to conclude: if you never heard of open sets, closed sets and closures before, then don't study topology but rather analysis. If you met them before in the study of $\mathbb{R}$ (or even better: in $\mathbb{R}^n$, or still better: in metric spaces), then you're ready for topology.

 

[homeomorphic]

Some people might do okay with topology first, but if you prefer to have some motivation sooner, rather than later, definitely analysis should come first. Or you could do both concurrently, maybe.

I have a strong preference for studying things in an order that makes them seem natural, rather than being pulled out of a hat, which often makes studying math very frustrating to me, since textbook authors often make a different choice for you.

 

[ahsanxr]

I see, thanks for the opinions guys. I think instead of jumping straight into diff top next year I may just do an independent study in Topology. Regardless I'll still be studying analysis over the summer.

 

[nucl34rgg]

So I'm planning to delve into both of these subjects in some depth during the summer to prepare for undergrad analysis (using rudin) and a graduate differential topology class. My question is which one should I start out with and pay more attention to. I obviously need to study a lot of topology if I am to comprehend that diff top class but I've heard conflicting views on which should be studied first, for example,

Micromass says:



whereas Monocles says:



So what are your opinions on this?

As for texts, I am looking at Munkres for Topology and Pugh for Analysis.

I think Munkres is a pretty gentle introduction to Topology. I am working through it now. I think the topology would help for analysis.

 

[zooxanthellae]

I'm not sure that I'd recommend Pugh for analysis. A lot of people like it, but I personally thought it was at times too informal. A more advanced student described it nicely by saying that he would not have liked to learn from it, but that he found it very useful for remembering concepts and intuition after the fact.

I like Kolmogorov and Fomin's "Introductory Real Analysis" a lot. It's not as chatty as Pugh but I didn't find it as terse and dry as Baby Rudin, either.

 

[homeomorphic]

I'm not sure that I'd recommend Pugh for analysis. A lot of people like it, but I personally thought it was at times too informal. A more advanced student described it nicely by saying that he would not have liked to learn from it, but that he found it very useful for remembering concepts and intuition after the fact.

From my point of view, that does sound like a recommendation, at least as a supplement. I like to have the intuition as soon as possible, not after the fact.

 

[zooxanthellae]

From my point of view, that does sound like a recommendation, at least as a supplement. I like to have the intuition as soon as possible, not after the fact.

I prefer a more rigorous approach that slowly yields intuition: that way I feel more confident in the way the material works. Some people like an intuition-based approach. If you're one of them, by all means, go for it.

 

[ahsanxr]

I'm not sure that I'd recommend Pugh for analysis. A lot of people like it, but I personally thought it was at times too informal. A more advanced student described it nicely by saying that he would not have liked to learn from it, but that he found it very useful for remembering concepts and intuition after the fact.

I like Kolmogorov and Fomin's "Introductory Real Analysis" a lot. It's not as chatty as Pugh but I didn't find it as terse and dry as Baby Rudin, either.

In the class I'll be taking, we'll be using Rudin, which I hear is terrible to learn from if it's your first exposure to analysis. So the point of studying over the summer would be to get some intuition and familiarity with the concepts before I start tackling Rudin. Then again, I don't want a text that's too introductory and informal like Abbot's "Understanding Analysis". So I think Pugh should work well.

 

[Robert1986]

In the class I'll be taking, we'll be using Rudin, which I hear is terrible to learn from if it's your first exposure to analysis. So the point of studying over the summer would be to get some intuition and familiarity with the concepts before I start tackling Rudin. Then again, I don't want a text that's too introductory and informal like Abbot's "Understanding Analysis". So I think Pugh should work well.

The book we use for undergrad topology is The Elements of Real Analysis by Robert Bartle. This is an interesting book, and a little different from the other analysis books I have seen. Metric Spaces aren't mentioned except for in one exercise. However, the first portion of the book is all about the topology of R^n, which is just a special case of more general topologies. So, it kind of let us get our feet wet with open sets, closed sets, convergence, etc.

It might be worth checking out if your library has it and working through the first part of the book. This is a rather expensive book, so I wouldn't recommend buying it unless you can get a really good deal. It is really something between an Advanced Calculus book and, say, Rudin (not baby Rudin, the other one.)


Now, are you using Baby Rudin (Introduction to Real Analysis, or something like that) or "Big" Rudin (Real and Complex Analysis.) If you are using the latter, I feel sorry for you. This is not a good introductory book.

 

[Monocles]

You need less topology than you probably think to start learning graduate-level differential topology. A lot of the material you learn in a first topology class won't be relevant, since every space you deal with in differential topology is very 'nice', without much intuition-violating behavior. It is when you have to start dealing with stranger topologies such as the ones you encounter in algebraic geometry that more of a first topology class will become relevant.

On the other hand, a lot of what you learn in analysis will be relevant in differential topology. That was my biggest weak point when I tried to do grad intro differential topology and undergrad intro analysis at the same time - for example, I ran into the much more general implicit function theorem in the case of smooth manifolds before the implicit function theorem in the case of R^n! I managed to barely get by with an A in the class but I got the impression that I was spending twice as much time on the material as the people who had the proper prerequisites. And now I am entering grad school in the fall and I plan on retaking differential topology since I didn't learn as much as I should have the last time around due to insufficient background. So there is a conclusion on the paragraph of mine that you quoted in the original post.

 

[homeomorphic]

I prefer a more rigorous approach that slowly yields intuition: that way I feel more confident in the way the material works.

This seems strange to me, since I tend to think that "intuition" is what tells you why it works, whereas logic tells you that it works, but not why. Plus, intuition is easier to remember. The intuition is the thing that the subject springs forth from. Like a seed from which it grows. That more closely resembles the way math is discovered, usually. Also, there's nothing stopping you from getting the intuition AND doing the rigorous proofs.