Analysis book that assumes some knowledge of topology

[jojo12345]

I realize this may be kind of strange, but as it turns out, I've experienced a good introduction to topology (at the level of Munkres) before taking a rigorous class on analysis. With a month-long winter break in my near future, I'm wondering if anyone could suggest a text on real analysis that assumes prior knowledge of topology so that I might study it independently of school.

 

[fluidistic]

I'm not 100% sure, but I think that this book assume some basic knowledge of Topology : http://press.princeton.edu/titles/8008.html.
It seems also good. On the other hand I'm just a physics (no math major) student and I don't own the book.

 

[letmeknow]

By rigorous class on analysis, which do you mean. I've encountered 2 classes that are called analysis (at my university it is 4 classes, 2 sequences).

There is the "analysis" which is proving the theorems from calculus which uses books like "Understanding Analysis" by Abbott and "Principles of Mathematical Analysis" by Rudin.


Then there is "real analysis", which studies borel sets, lebesgue integrals, lebesgue measure, etc. Books like "Real analysis" by Roydin and "Real and Complex Analysis" by Rudin.

My understanding is that if you put the word "real" before the course, you are talking about the second level of analysis which is harder.

 

[fourier jr]

the think the op was about an intro book, not measure theory

 

[letmeknow]

the think the op was about an intro book, not measure theory

Then I'm not so sure that any book will assume Topology. Even in Rudin's Intro treatment of analysis he devotes 1 of 9 chapters to Topology (Rudin's treatment is known to be difficult for most).

Choose any book that has a chapter on topology in it and start working through it. If the topology section is trivial, then skip it. But make sure to pick a book that uses topology because some don't...and it's thought to be better if you do use topology.

 

[jojo12345]

I've actually got the second of the two Rudin books you mentioned sitting on my hard drive. I skimmed the first few sections of the first chapter on measure theory, and while I don't find this material to be over my head, I think I'm looking for the material in the other of his books you mentioned, the one that concerns itself with proving things about calculus.

Here, I'll be more clear about the math background I have. As of a year ago, I've taken the engineering courses at cornell on single variable calculus, multivariable calculus, differential equations, and linear algebra. These were all taught by math faculty, but they were geared toward providing us engineers with a working knowledge of this stuff. So, these classes sacrificed some rigor for the sake of covering more material (except for linear algebra, which was nice).

Currently I'm near the tail end of a course on topology taught by the math department for math majors. The first 2/3 of the class was dedicated to getting a hold on such concepts as compactness, connectedness, quotient spaces, etc.. Now, we're doing some introductory algebraic topology: defining the fundamental group, covering spaces, classification of covering spaces, etc.. The class has been great fun for me and, as a result, I'm keen on learning a good deal more about mathematics.

Seeing as I'm currently a senior applying to graduate schools for plasma physics, doing this will involve me reading textbooks on my own. As most math books assume the reader has experienced some sort of introductory exposure to analysis, I feel it would be appropriate for me to gain just this sort of exposure. Furthermore, because I already know some topology, it would be expedient to learn this material from a text that doesn't avoid using topology.

Do you have any other recommendations for texts with a chapter on topology early on, besides Rudin?

 

[fourier jr]

I've actually got the second of the two Rudin books you mentioned sitting on my hard drive. I skimmed the first few sections of the first chapter on measure theory, and while I don't find this material to be over my head, I think I'm looking for the material in the other of his books you mentioned, the one that concerns itself with proving things about calculus.

ha! I knew it! :p since intro analysis seem to be a standard course, there are lots of books to choose from. I learned from the one by pfaffenberger/johnsonbaugh, which has ~750 problems. principles of mathematical analysis by rudin covers similar material was sort of the original one, written more than 60 yrs ago but people say it isn't very user friendly. I've never had a good look at the ones by apostol, ross, or pugh but they seem to be popular also. usual topics in those books include the theorems from calculus, but in a general metric space, rather than a metric space with the euclidean metric, forms of the baire category theorem, heine-borel theorem, maybe 3d-calculus, sequences & series incl Fourier series... i think they'd all have some topology in them because that's the course where those sorts of concepts (open & closed sets, compactness, completeness, etc) are usually introduced.