Next Steps in Math: Analysis to Topology or Real Analysis?

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The discussion centers on transitioning from Stephen Abbott's "Understanding Analysis" to subsequent mathematical studies. After completing Abbott's book, there are several recommended paths. One option is to delve into topology, which expands on concepts like convergence and continuity, though it's suggested to first study metric spaces since Abbott may not cover them extensively. Another recommendation is to pursue advanced real analysis, particularly focusing on Lebesgue integration, with "Principles of Real Analysis" by Aliprantis and Burkinshaw highlighted as a strong choice. Additionally, functional analysis is mentioned as a viable option, with Kreyszig's book noted for its accessibility. The participant expresses interest in exploring different types of integration and is considering advanced analysis while still completing Abbott's work. There is also a request for specific topology book recommendations, indicating some confusion about the various types available.
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Recently, I started learning Analysis from the Stephen Abbott Understanding Analysis book, and I feel like I can finish it within the next month. So I was wondering, what other book should I study after this one? With an intro to analysis, would I be ready for an intro to Topology? If so, which book should I study with?

Or should I perhaps get a more advanced book on Real Analysis?

What do you guys think?
 
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There are several things you can do now. There isn't really a "fixed path" on what you need to take.

Some ideas:

1) Topology: this will generalize ideas like convergence and continuity. The topology is actually the study of open sets of a certain space. I don't think Abbott treats metric spaces, so I suggest you study some metric spaces first.

2) Advanced real analysis: this will likely introduce Lebesgue integration. Riemann integration is handicapped in several ways, Lebesgue integration fixes this. "Principles of real analysis" by Aliprantis and Burkinshaw is a very good book.

3) Functional analysis. The book by Kreyszig requires no real knowledge of topology or advanced analysis.. It will introduce basic functional analysis: Hilbert spaces, Banach spaces, operator algebras, etc. This book is a very good book to study after Abbott.
 
micromass said:
1) Topology: this will generalize ideas like convergence and continuity. The topology is actually the study of open sets of a certain space. I don't think Abbott treats metric spaces, so I suggest you study some metric spaces first.
.

It did mention mention metric spaces briefly after proving what I'm guessing is a special case of the Baire Category Theorem, that R is not the countable union of no where dense sets. I'm about half way through though, so I don't know if I will ever see metric spaces again in that book.

Would you recommend any Topology book in particular? I've been looking online, but I see Point set something topology, algebraic topology, general, etc, etc. And I have no idea which one is the one for me you know?

For now, I think I will go with the advanced analysis one you mentioned, though I still have to finish the Abbott one. Since I read somewhere that there exist other types of integration, I've been very eager to learn more, and I might finally understand what the wiki article talked about... lol.

Thanks.
 
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