Should I take modern(abstract) algebra or topology first?

[tyrrhenus]

Hi, I am trying to decide whether I should take a modern algebra or topology course next semester. I have a bachelor's in physics but I have not taken very many higher math classes. This is a list of the relevant classes I have taken.

Calculus (up through partial differential equations)
elementary linear algebra (not very theoretical)
geometry (loaded with proofs, senior class)
Set theory and logic

Set theory and logic, and geometry were two of my favorite class in college, and I did very well in geometry and would have done very well in set theory and logic if I had not been taking so many classes then. I am going to apply for a master's/phd in math but I need to take several classes by correspondence before I do so. Right now I could either take modern algebra or topology first, but I am not really sure which one to take first. If it is any help I am going to take classes through UCR (University of California, Riverside).

Also if anyone has a good suggestion on a book that will help me to get a head start I would appreciate the suggestion. I have been studying my calculus book from college the last couple of months but want to review more of the higher math stuff. A few books I have found are.

Set Theory and Logic by Robert R. Stoll
Elements of Logic via Numbers and Sets by Johnson (Springer Undergraduate Mathematics Series)
Introductory Mathematics: Algebra and Analysis by Geoffrey C. Smith

I of course have heard about Spivok's Calculus and "Baby Rudin" but I thought I would put those off a couple of months.

 

[micromass]

It doesn't really matter very much. Undergraduate classes in topology and algebra are usually very distinct, so choose whatever you like.

From a physics point of view, take topology. It will be much more useful than abstract algebra.
If you like set theory and geometry, then you'll adore topology!

For a head start: read Munkres' topology book.

 

[qspeechc]

Maybe you should study more linear algebra? It's very important to both mathematics and physics. There's the book by Shilov. For topology a much cheaper book than Munkres is Willard, though it is quite challenging, or some of the other Dover topology books, which are apparently very good. Also there's no reason not to read a book like Spivak right now, which should be good preparation for higher mathematics, particularly analysis, geometry and topology. My opinion is that topology makes much more sense if you know some analysis, such as in Spivak. Like micromass said, algebra and (general) topology are independent.

 

[tyrrhenus]

Thanks, that really helps me. I will probably take topology first if the order doesn't really matter.

 

[Matrix0]

I would recommend taking topology first before modern algebra. Both subjects are important and useful in mathematics, but topology has more practical applications in physics, especially in fields such as quantum mechanics and general relativity. Topology is also a more visual and intuitive subject, which may make it easier to grasp for someone with a background in physics.

In terms of books, I would suggest "Introduction to Topology" by Bert Mendelson as a good starting point. It covers the basics of point-set topology and has a lot of examples and exercises to help with understanding. "Understanding Analysis" by Stephen Abbott is also a great book for reviewing higher level math topics such as analysis and proofs.

Ultimately, the decision of which course to take first depends on your personal interests and goals. If you are more interested in abstract concepts and algebraic structures, then modern algebra may be a better option. But if you want to build a strong foundation in mathematical analysis and its applications, then topology may be a better choice. Good luck with your studies!