Should I Take Topology in the Spring?

[Chaostamer]

Hey everybody. I'm a Pure Math major and I'm trying to finalize my schedule for next semester. Originally I was enrolled in a second ODEs course (focusing predominantly on systems of linear ODEs, existence and uniqueness theory, and qualitative solutions), but after a rough semester with PDEs, I need a break from the more "applied" classes. The issue has become finding a class from my rather limited set of options to take its place.

I'm most interested in taking Topology, but I have a few concerns about doing so. Among them is my background. Among my math classes, I've taken an introductory, proof-based course in discrete mathematics and a first course in linear algebra that put a fair bit of emphasis on theory. This semester, I also took courses in Combinatorics and Theory of Computation that focused largely on proofs. The point is, I definitely have some proof experience, and in those classes, I did pretty well.

However, I haven't taken any courses in analysis or abstract algebra (and won't be able to until the Fall semester). I was told that both of those classes make topology a good bit easier to understand. Plus, since Topology is very rigorous, do you think I have the background to handle the material and appreciate it? I don't want to take it if I'm going to be lost the entire semester or if I lack the perspective to appreciate the subject.

Also, how much will it benefit me to take Topology before analysis. I know the two go hand-in-hand quite often and I was hoping that, should I take Topology, it would provide more motivation for my year of analysis. Also, if I do take it, it would provide me with more practice with rigorous proofs. Is the benefit worth it?

Okay, so I got a bit long-winded here trying to sort out my thoughts, but I'd like to hear your perspective on the matter. For reference, these are the other math classes I'm taking next semester:

Linear Algebra II
Graph Theory
Statistical Theory II (advanced probability theory and introductory mathematical statistics)

Thanks in advance for your help.

 

[Chaostamer]

Oh, I should clarify that analysis isn't a prerequisite for topology (the undergrad topology course at my school provides the necessary information on metric spaces and real numbers), nor is topology needed for analysis. Theoretically I can take the two in either order; my hope is that whichever subject I study first will motivate the second.

I should also mention that I have books on both subjects already and have been reading into them over my break. I at least have some idea of what topology entails.

 

[capandbells]

What year are you? You might be able to appreciate topology more if you take analysis first: if anything, analysis will provide motivation for studying topology. Having said that, there's no reason you couldn't, having already had a substantial introduction to proof-writing, take a topology course without having taken analysis. I would say go for it.

 

[axiomatic7777]

Or take analysis and topology at the same time. It's a bit intense, but it will surely reinforce your command of the materials that are common to both.

 

[Chaostamer]

Wow, you guys are fast. Many thanks.

capandbells, I'm in my third year now. I understand Analysis will motivate Topology, but will Topology provide any significant motivation for Analysis?

axiomatic, unfortunately, the first course in Analysis is only offered in the Fall while Topology is only offered in the Spring. The best I can do is take my second Analysis course along with Topology, but that involves putting off Topology another year.

 

[╔(σ_σ)╝]

I guess it's up to you. Since analysis is not a preq you should be fine. But I myself would not feel comfortable in a topology class without analysis.

To some extent, analysis makes you less naive and more able to "see" things.

 

[Monocles]

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.

 

[Chaostamer]

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.

See, that's what I was hoping to hear. In my undergrad program, the hardest math classes are generally considered to be Abstract Algebra, Analysis, and Topology, with either of the latter two generally considered the most difficult. I'm taking Analysis and Algebra in the fall, along with a couple graduate-level courses, so it's going to be a pretty rigorous semester. Meanwhile, my other spring courses are going to be challenging, but something of a lower level of challenge (Linear Algebra and Graph Theory are proof-based, but they're a good bit more intuitive than the other upper-level math classes and Stat Theory II has a fair number of computational problems).

Anyway, my hope was that if I took Topology in an "easier" semester, I'd get more practice with a high level of mathematical rigor and help make my year-long Analysis sequence more understandable.

My other question is about the textbook. I know Munkres is considered the standard for undergrad Topology. (I've personally been self-learning from Kasriel over the last couple weeks.) The professor, however, has chosen C. Wayne Patty's Foundations of Topology. I skimmed the first chapter (on metric and topological spaces) earlier this week and it seems well-written, but does anyone out there have any experience with it?

 

[Troponin]

Well, this thread has me feeling even more anxious about next semester than I already was...
I'm a physics major who is taking topology next semester...who also hasn't had analysis or abstract algebra.
I'm hoping my interest and prior self-study in differential geometry (for general relativity) will be enough to allow me to keep from getting too lost...and possibly even come away from it with a better "base" to continue on in diff. geometry and GR.

I don't really have anything of substance to add...just wanted to chime in and wish you good luck in the class.

Maybe you could give an "update" post sometime during the semester?
I know we get a number of posts like this (usually for similar classes), so it would be nice to see how things end up for one of the students asking the question.

 

[Chaostamer]

Well, this thread has me feeling even more anxious about next semester than I already was...
I'm a physics major who is taking topology next semester...who also hasn't had analysis or abstract algebra.
I'm hoping my interest and prior self-study in differential geometry (for general relativity) will be enough to allow me to keep from getting too lost...and possibly even come away from it with a better "base" to continue on in diff. geometry and GR.

I don't really have anything of substance to add...just wanted to chime in and wish you good luck in the class.

Maybe you could give an "update" post sometime during the semester?
I know we get a number of posts like this (usually for similar classes), so it would be nice to see how things end up for one of the students asking the question.

I emailed the professor to ask him about my concerns and he said, "Super students will do well regardless of background". I think it's just going to come down to work ethic for me. Hopefully the situation is similar for you.

I'd be glad to give an update. This seems like a pretty great forum anyway, so I'll probably be sticking around.

 

[eumyang]

However, I haven't taken any courses in analysis or abstract algebra (and won't be able to until the Fall semester). I was told that both of those classes make topology a good bit easier to understand. Plus, since Topology is very rigorous, do you think I have the background to handle the material and appreciate it? I don't want to take it if I'm going to be lost the entire semester or if I lack the perspective to appreciate the subject.

Too late to mention, but I find it odd that you are not taking Analysis right now. At the school I went as an undergrad, all math majors were required to take Analysis I the fall semester of junior year, regardless of concentration. Maybe it's just my school. I wonder if there other schools where Analysis is required/recommended earlier... or even later.

On the other hand, my school didn't offer a Topology class at all when I was there. I've heard that since then, however, they offer it sometimes as a special topics course.

 

[Fredrik]

will Topology provide any significant motivation for Analysis?

Not really, but it will make it a lot easier for you to understand analysis. (Many theorems in analysis are really just theorems in topology applied to functions from ℝ into ℝ).

 

[Chaostamer]

Too late to mention, but I find it odd that you are not taking Analysis right now. At the school I went as an undergrad, all math majors were required to take Analysis I the fall semester of junior year, regardless of concentration. Maybe it's just my school. I wonder if there other schools where Analysis is required/recommended earlier... or even later.

On the other hand, my school didn't offer a Topology class at all when I was there. I've heard that since then, however, they offer it sometimes as a special topics course.

My school recommends Analysis in Senior year. However, that assumes that Math majors begin with Calculus I. Those who have more math experience going in can generally get through ODEs, PDEs, and a sequence of Linear Algebra within their first two years, thereby allowing them to take Analysis a year earlier--and I know some who take it as early as their second Fall.

Regardless, though I'm a third-year now, I started undeclared and then moved to Computer Science, so I got a bit of a late start with my Math major. I could've taken Analysis this Fall, but I was afraid that, when I registered, I didn't have the experience to succeed. Obviously, my last couple semesters have made me a good bit more confident.

 

[Chaostamer]

Another question: how much fundamental analysis do I need to know for Topology? Kasriel's book starts out with some review of the structure of ℝ and some relevant theorems before it gets into metric spaces. How much of that ought I read up on before I start with the metric spaces?

 

[uman]

I took topology before analysis. In my experience the students who had already taken analysis (about 75% of the class) tended to be at a disadvantage because they found it difficult to avoid using theorems or facts that are true in R^n but not true in general topological spaces.

 

[Chaostamer]

I took topology before analysis. In my experience the students who had already taken analysis (about 75% of the class) tended to be at a disadvantage because they found it difficult to avoid using theorems or facts that are true in R^n but not true in general topological spaces.

So you think I might be better off doing it this way?

 

[uman]

Not necessarily. Taking analysis first has a lot of advantages. I'm just pointing out that taking topology first has advantages too.

However, if you've never had any proof-based class, you'll probably flounder in topology more so than you would in analysis.

 

[Chaostamer]

Not necessarily. Taking analysis first has a lot of advantages. I'm just pointing out that taking topology first has advantages too.

However, if you've never had any proof-based class, you'll probably flounder in topology more so than you would in analysis.

Well, I've taken a handful of proof-based classes.

Discrete Mathematics: All proofs, albeit at an introductory level.
Linear Algebra: A first course--about 50% theory.
Combinatorics: About 60-70% proofs, give or take depending on the specific section.
Theory of Computation: All proofs, but many of them were "high-level" CS proofs.

I haven't yet taken an intensively rigorous Linear Algebra course (I'm taking that this Spring) or the aforementioned Analysis or Abstract Algebra. However, I have been reading up on Topology and plan to have covered at least the basics (and whatever Analysis I might need) before next semester starts.

From your experiences, do you think I have sufficient background?

 

[uman]

Absolutely.

Please remember NOT to get complacent and stop going to class or studying regularly if the material is easy at first... This is how I got lots of Bs and one C this semester.

 

[Chaostamer]

Absolutely.

Please remember NOT to get complacent and stop going to class or studying regularly if the material is easy at first... This is how I got lots of Bs and one C this semester.

I didn't have the best semester this time either, but I realized that my performance is pretty directly linked to my interest in the material and I had a hard time getting into PDEs. Next semester should be a lot more interesting and I plan to put a lot of effort into my classes--Topology in particular. Fortunately, it's my only class on Monday and Wednesday (the time in which it's offered), so I'll have most of the day to make sure I'm completely on top of it. I never skip classes either, so that helps a good bit.

I really appreciate the advice and encouragement, though. This sounds like a pretty challenging path, but I think it could be worth it in the long run.

 

[Monocles]

Another question: how much fundamental analysis do I need to know for Topology? Kasriel's book starts out with some review of the structure of ℝ and some relevant theorems before it gets into metric spaces. How much of that ought I read up on before I start with the metric spaces?

Beyond being familiar with the notion of cardinality of infinite sets, I didn't know anything about analysis when I started with topology. If the textbook is self-contained enough (like Munkres) then you shouldn't need to know any.

 

[Chaostamer]

I don't really have anything of substance to add...just wanted to chime in and wish you good luck in the class.

Maybe you could give an "update" post sometime during the semester?
I know we get a number of posts like this (usually for similar classes), so it would be nice to see how things end up for one of the students asking the question.

So, here's my requested update.

I got through Topology and managed to pull through with a 'B', so I'm pretty content. The course was super-challenging. Turns out, I managed to get basically the most difficult professor I could possibly get. He was really rigorous and honestly, not too generous with partial credit on the exams. He also moved a lot faster than professors tend to do in that class, so it took a lot of work just staying on pace.

That said, he was an incredible lecturer and the fast pace of the course meant that we were able to cover a lot of topics that, apparently, introductory topology courses don't cover (he even started lecturing out of a Bourbaki book in the last month). By the end, we were doing a good bit of work with compactification theorems, topological groups, function spaces, and geometric interpretations. I didn't absorb all of it as well as I would've liked, but I think I'll benefit from having seen the material.

Overall, I'm pretty glad I took it this semester. The downsides are that I put work into Topology at the expense of my other classes, so I ended up with a 'B' in both Linear Algebra and Graph Theory, the former of which (at least) should of been a definite 'A'. I also feel like I got more of a superficial feel for the subject than an in-depth intuition for it. I'll probably have to retake the course at some point in grad school to really be comfortable with it, but I really, really enjoyed it, so I'm pretty okay with that. After getting through this course, I've found it a lot easier to manage rigorous subjects, so my independent reading of analysis and algebra has gone much more smoothly.

So, thanks to everyone who gave me advice and wished me well before the semester. I'd say it was a pretty positive experience.