Analysis with or without set-theoretic topology?

[inthenickoftime]

Do you think a first course in analysis should focus entirely on inequalities and leave set-theoretic topology for another occasion? Should this depend on whether or not the student had a first rigorous calculus course first? If I'm not mistaken, Victor Bryant (Yet Another Introduction to Analysis) and Arthur Mattuck (Introduction to Analysis) authored analysis books in the language of inequalities. Has anyone had previous experience with these two? Did it alleviate topological proofs in later courses? Is topology even required for the level I'm aiming at? Let's say my main concern is grasping the calculus of variations (much needed in mechanics). At what point do you absolutely need to incorporate topology in your analysis?

edit: I'd like to add that David Bressoud's A Radical Approach to Analysis also does this.

 

[S.G. Janssens]

Do you think a first course in analysis should focus entirely on inequalities and leave set-theoretic topology for another occasion?

I think that a first course should do both: teach basic techniques in estimation, but also introduce core concepts of set-theoretic topology. At a more advanced level, these often go hand in hand.

Should this depend on whether or not the student had a first rigorous calculus course first?

No, as part of the purpose of a first course in analysis is to make calculus rigorous.

Victor Bryant (Yet Another Introduction to Analysis) and Arthur Mattuck (Introduction to Analysis) authored analysis books in the language of inequalities. Has anyone had previous experience with these two?

No, sorry, I have no experience with those texts.

Is topology even required for the level I'm aiming at? Let's say my main concern is grasping the calculus of variations (much needed in mechanics). At what point do you absolutely need to incorporate topology in your analysis?

It really depends on what level of understanding you are striving for, and what you aim to do with the theory that you will learn. For example, a lemma by Weierstrass says that a continuous real-valued function (i.e. a continuous functional) on a compact topological space assumes its maximum and minimum. Now, calculus of variations is typically concerned with the case that the domain of the functional is a subset of an infinite dimensional topological vector space. In this setting, it is typically not easy for that domain to be compact, and both compactness and continuity depend very much on the topology in question.

I'm quite sure there are lots of physicists that use calculus of variations successfully (for example in mechanics) without worrying about topological issues. On the other hand, it are precisely those issues that I find more interesting myself.

So, my advice would be to get a rigorous introduction to analysis that offers a broad perspective and does not limit itself to one very specific approach. You can always study such approaches in tandem or (maybe better) after having finished your introduction.

 

[mathwonk]

I am not sure of the answer to your question, but I have some experience. In my own math career I have found that basic topology is probably the most useful and fundamental language and knowledge there is, absolutely valuable to almost everyone, certainly to me. The introduction to John Kelley's General Topology said he was dissuaded with difficulty by friends from titling his book "What every young analyst should know". So in my opinion you will be glad for everything you learn from topology.

 

Maybe I'm biased, but one cannot know too much topology. It is such a powerful tool. I use it all the time.

 

[S.G. Janssens]

Topology is beautiful and useful, but for an introduction to analysis, I think it is best to find a balance between "soft" and "hard" analysis.

 

[inthenickoftime]

Topology is beautiful and useful, but for an introduction to analysis, I think it is best to find a balance between "soft" and "hard" analysis.

I had to ask because topology and sets are relatively new. If I'm correct, people were doing rigorous mathematics long before these two branches were introduced (I can think of Fourier and Cauchy). The addition of new notions (propositional logic, sets, topology, etc.) to an already complex subject (calculus) when others have done without, and, successfully at that, made me wonder if the actual math curriculum isn't doing its students some disservice by obfuscating elements of calculus with the introduction of other topics prematurely. I'm not arguing against their use, but question their use within a first rigorous course on calculus.