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$\quad$ Hello everybody, I have a bit of a problem and I know this question has been discussed before but I just want some insight on how to do well in more higher level math courses (There is a $\textbf{TL;DR}$ version of this at the bottom if you don't feel like reading a block of text). I am currently taking an introduction to topology course (we are using Munkres as the text) and I find myself doing rather poorly for now. I enjoy doing proofs more than computation but I still find my proof abilities stagnant. All of our assignments have literally been questions from the textbook and I often do not know how to even approach the proofs. Sometimes I will write down the given information and maybe a definition or two but it still boggles my mind at times how to solve these questions.

In terms of my background in math I have never taken an analysis course, I took a single-variable calculus course and a course called Advanced Calculus (which to be honest was actually very computational and very easy, the most theoretical idea we learned was how to do epsilon-delta proofs for multi-variable functions, but no euclidean topology, which is apparently kind of common for the course I took but for some reason when I took it they cut a lot of more theoretical stuff out). I have taken other more proof-heavy courses including a courses in linear algebra, group theory, ring and polynomial theory, number theory, and differential geometry (which was an odd course because the textbook questions were often quite theoretical but our midterms and exam was mostly computational, like finding the first and second fundamental form or calculating Gaussian curvature and stuff like that, so I ended up doing really well since I do well in courses that are just computation. So although topology is not my first foray into proof heavy courses this one is certainly my most difficult and the most rigorous course I have ever taken. I really don't know how to do well in this course, or at least I don't think I know how to do well. Often I just end up finding the solutions for questions cause I just get too frustrated (I do not do this typically when working on assignments though... I really hate the feeling of possibly cheating). So, I guess the question really is the approach to courses like this because I really do want to take more higher-level courses cause I find all this stuff super fascinating (I am a math major so I really am serious about this) but I just feel overwhelmed and stressed a lot of the time. Thank you for any answers.

$\textbf{TL;DR}$ I am a math major who has continually struggled in more abstract and proof-intensive courses (specifically right now a first topology course that I am taking) and I would just like some advice on how to improve in proofs and just math in general!

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1 Answer 1

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I suspect your problem with topology is that, without the usual progression of sequences and series, real and complex analysis, metric spaces and then point-set topology, you've not had the chance to develop your intuition on how to go about attacking a problem in the simpler settings. For example, topological proofs tend to involve a lot of looking at the images/inverse images of sets, which can be a bit confusing, especially when you haven't seen similar proofs in the case of metric spaces.

Just for the sake of building up a few pictures that you can use to try and get the idea of what's happening in a proof, it might be a good idea to learn some basic metric spaces - at least enough to understand the idea of where the definitions of an open set and continuous function which are then used in the general setting of topological spaces come from. When I was learning metric and topological spaces, I found Sutherland's book to be pretty useful. Not only is it a fairly cheap book that serves as a good introduction to metric spaces, but it will hopefully help you out with some topological ideas too.

Once you've got a very basic idea of metric spaces and appreciate what's going on in the metric space proofs (i.e. choosing small $\epsilon$-balls), you'll be able to draw pictures to help you see what's going on, and see if you can generalise that to topological spaces (so replacing balls with open sets). Drawing pictures is in general a good way to get the hang of definitions and theorems in topology. Even if it's completely unrigorous, sitting and thinking about what a definition is saying and trying to draw a picture is always a good way to build up an intuition for what it's actually saying.

As far as general proofs go, you certainly get better with practice. A lot of the people I know that find proofs difficult aren't so much struggling with the maths behind it, but rather struggling with the idea of what exactly a proof involves, especially as simpler metric spaces / topology proofs are often just combining and rewriting definitions. Being clear in how you set out the start of your proof is always a good idea. You want to show that a set of assumptions leads to a conclusion, so set it out along the lines of:

Suppose [assumption 1], [assumption 2]. That is

-definition of what [assumption 1] actually means, i.e. if we have assumed a function is continuous, the inverse image of an open set in its image is open.

-definition of what [assumption 2] actually means.

Then we want to prove [conclusion], i.e. definition of what [conclusion] actually means.

Quite often, when you've got the definitions spelled out in front of you, something will jump out as a good starting point. Obviously at this point, I can't keep giving general tips as not all proofs are the same. The one important thing is to always keep in mind what you're trying to prove, i.e. what your conclusion is. If you always know exactly where you want to go, then there's usually going to be an obvious next step from what you have in front of you. And as long as you're careful about writing out exactly what you mean at each step, you'll always have everything you know in front of you.

Oh and while I'm at it, another book that I remember being useful when I was a first year undergraduate is this one. The content is rather basic, and none of the proofs are difficult at all, but it's good in that it explains what the proofs are doing very clearly, and often preceeds an actual proof with a short discussion of how to choose the correct strategy for that proof.

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Thank you @Peter. That was the type of in-depth answer I was looking for. I will definitely look into those books. I also occasionally look at Topology Without Tears by Sydney Morris, which I find is slightly easier to handle then Munkres. In any case thanks for the advice :) –  InsigMath Oct 26 '12 at 6:13

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