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

[00Donut]

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?

 

[micromass]

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.

 

[00Donut]

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.