# Topology vs more Analysis

I'm entering into a graduate statistics program in the coming year and don't really need either class for my Master's. However, I am considering applying for a Ph.D in mathematics in the future, but for now I want to take an elective math course for fun. I've already taken a year of Real Analysis as an undergrad and LOVED it, but I've never taken Topology. I was wondering what you guys think I should do: (a) take more analysis and continue learning more of what I already know I love or (b) take topology and explore new territory.

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Woops! Didn't mean to post here. Can this be moved to the Academic Guidance forum?

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I'm kind of surprised that your graduate statistics program does not require real analysis since basic analysis is used quite often in basic limiting theorems - say distributions whose limits are Poisson or informal derivations of the central limit theorem - while graduate real analysis lays the foundations for probability theory.

Anyways I think it all depends on what your undergrad course consisted of. If you used something like Rudin, you probably already have a very good idea of basic topology (since one of the chapter titles is called just that) in metric spaces. In this case I would recommend a more advanced analysis course so you can see how to apply basic analysis techniques to some very interesting theory. I'm imagining some course that involves functional analysis, measure and Lebesgue integration, or fourier analysis.

If you haven't been exposed to metric topology, then not surprisingly I recommend topology. A lot of what you'll be doing will feel like analysis, except you won't be explicitly working with a metric, or a distance, but with open sets. A simple reason for this is that topological considerations will underlie many topics in analysis, and topology started out by generalizing the notions of limit and distance in basic real analysis.

Not sure if this is particularly sensical. I've learned a healthy amount of real analysis, but I only know the basics of topology so I've tried to make things unbiased :P.

Statistics? Then for advanced studies you need measure theory and functional analysis. If these courses are already in your graduate program, I would take a topology course now. It'll help a lot in understanding how things are abstracted in modern math from concrete structures. Plus topology is everywhere in modern math.