So, I'm just kind of curious as to why people recommend topology as a class to take in preparation for grad school. I'm currently taking it because:
1. I needed another elective for my math major
2. It looked interesting
3. Forums, department websites, econphd.net, etc. said it would be useful
So, how does it apply? Is it just that it gives one more experience dealing with rigorous proofs? Or does some of the actual material get used?
I believe it is very useful when you get further along in econometrics, but I haven't taken grad metric yet so that is just what I have heard.
The rigorousness of it doesn't hurt either.
If you find it fun you are in good shape (o man that can be a pun!)
It helps in econometric theory.
That said, I asked a handful of graduate advisors at top20s whether it would be better to take a course in General Topology, or Real Analysis II (at the Roydin level).
The response was overwhelmingly in favor of Real Analysis II, because it is something that is helpful immediately upon entry, whereas you can pick up the theory behind topological spaces in the actual PhD Econometrics series (three courses, sometimes four depending on the program and your speciality).
Solid. I'm taking Analysis using Rudin right now and was planning on taking RA II next semester anyway. Thanks for the feedback.
It really depends on what you cover on the second part of analysis. For the most part, the things that you need to learn are covered in the first part of analysis (e.g., intro topology, cont., differentiability in R^n, Reiman-Stl., etc...). Topology generalizes the idea in analysis. In fact, much of the topics in topology were motivated by problems in analysis. Topology, according to my professor, is the language of modern mathematics. That being said, it is hard to advance in Math without exposure to topology. Modern analysis and algebra, for example, use results from topology.
Originally Posted by butler blue
Keep in mind that at most schools, there is a course series called Analysis I & II (sometimes Advanced Calculus) that typically uses Rudin's "Principles of Mathematical Analysis".
A lot of schools have a graduate course series Real Analysis I & II subsequent to that based on Royden's "Real Analysis".
I'm attempting to get all the way through the latter, be it overkill or not.
Analysis not just uses results from topology, it is based on (general) topology. What do you think a continuous function is? And any advanced math class is mostly a training in proofs for non math major as presented material is usually way more general than what one might need in applications.
What you can do
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