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Analysis with or without set-theoretic topology?
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[QUOTE="S.G. Janssens, post: 6394796, member: 571630"] 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. No, as part of the purpose of a first course in analysis is to make calculus rigorous. No, sorry, I have no experience with those texts. 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 [I]functional[/I]) 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. [/QUOTE]
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Analysis with or without set-theoretic topology?
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