Favourite Dover Books in Analysis, Algebra, and Topology?

[Chaostamer]

I have a friend who, like me, is a Math major, although she started later than I did and as such, hasn't yet gotten into the core classes for her degree. She's frequently checked out my own personal library and I figured that, since the holidays are coming up, it might be cool to start her off with a small collection of her own.

So, basically, I want to get her three books, one for each of the core branches of math she'll be studying over the next year and a half. However, I want to keep it affordable, so I'm probably most interested in finding her good Dover books on analysis, algebra, and topology (although if there are good non-Dover books for around ten dollars apiece, I'm open to those suggestions as well).

Do you guys have any helpful suggestions for me?

[deluks917]

I don't know how strong/dedicated she is but Jacobson Basic Algebra is really good but very hard. On the other hand Bert Mendelson Topology is also goodish but a little easy. Rosenscilt intro to analysis is nice. Cartan has a nice book on complex variables "Elementary theory of analytic functions n one and several variables." All these books can be found for about 10 dollars.

[fourier jr]

there's a big list here
http://www.physicsforums.com/showthread.php?t=212220

[xristy]

Three Dover books that I like very much are:

1) Hocking & Young - Topology
2) Steen - Counterexamples in Topology
3) Edwards - Riemann's Zeta Function

[Sankaku]

Pinter: A Book of Abstract Algebra
http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
Brilliant Introduction to the Subject.

Jacobson: Basic Algebra I
http://www.amazon.com/Basic-Algebra-Second-Dover-Mathematics/dp/0486471896
Mentioned above, very good, but not very suitable as an introductory text.

Gelbaum & Olmsted: Counterexamples in Analysis
http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753
The best supplement you could have.

Unfortunately, I don't think any of the basic level Dover analysis books I have found can compare to Bartle. It is a bit more expensive than a Dover, but you can find international editions.
Bartle: The Elements of Real Analysis, 2nd Edition
http://www.amazon.com/Elements-Real-Analysis-Robert-Bartle/dp/047105464X
http://www.biblio.com/9780471054641

EDIT: I forgot this. I haven't read it, but I have heard good things about it. It might be a worthy replacement for Bartle.
Johnsonbaugh & Pfaffenberger: Foundations of Mathematical Analysis
http://www.amazon.com/Foundations-Mathematical-Analysis-Dover-Mathematics/dp/0486477665

Mendelson: Introduction to Topology
http://www.amazon.com/Introduction-Topology-Third-Dover-Mathematics/dp/0486663523
Pretty decent.

Willard: General Topology
http://www.amazon.com/General-Topology-Dover-Books-Mathematics/dp/0486434796
This is a classic but, again, it might not be suitable as an introductory text.

[mathwonk]

I would suggest reading amazon reviews of books before deciding, to get a better idea of them.

I also love cartan's book on complex analysis, somehow both clear and condensed, and jacobson's book on algebra, with the caveat it is probably not easy for beginners to read.

it is easier to recommend good dover books on slightly less basic topics than you asked for.

I recommend any book by andrew wallace on topology, but his books are on differential and algebraic topology, not basic metric space theory and so on.

That said, I suggest looking at his "introduction to algebraic topology", which begins with an introduction to the fundamental ideas of topology.

I also like the writing of Georgi Shilov, and suggest his elementary real and complex analysis. I also like his linear algebra.

I recommend in general easier more readable books than harder ones, since the idea here is probably to help the person get started, rather than provide reference books only useful after learning the topic.