# Is this study plan sufficiently general, or overly specialized? [closed]

My current study plan is in order below. I will be completing these textbooks in this order one at a time.

I have been told that I don't have textbooks in my plan that approach topology in a general matter, but instead my focus is very narrow, meaning I may miss many important ideas.

Question: Is this a valid concern? Is the same true for either Algebra, or Analysis?

• Cohn - Classic Algebra
• Rudin - Principles of Mathematical Analysis
• Lee - Topological Manifolds
• Cohn - Basic Algebra
• Rudin - Complex Analysis
• Lee - Smooth Manifolds
• Cohn - Further Algebra
• Rudin - Functional Analysis
• Lee - Riemannian Manifolds

Note: My calculus is sufficiently built. For meta discussion on the close votes, please go here.

• Cohn – Classic Algebra
• Axler – Linear Algebra Done Right
• Zorich – Mathematical Analysis I
• Rudin – Principles of Mathematical analysis
• Dugundji - Topology
• Lee – Topological Manifolds
• Zorich – Mathematical Analysis II
• Rudin – Real & Complex Analysis
• Cohn – Basic Algebra
• J.P May – A Concise Course in Algebraic Topology
• Ddo Carmo - Differential Geometry of Curves and Surfaces
• Lee – Smooth Manifolds
• Cohn – Further Algebra
• Rudin – Functional Analysis
• Lee – Riemannian Manifolds
• @Committingtoachallenge Lee's books would be more accurately described as Differential Geometry texts. For Topology, I recommend Dugundji - Topology (after Baby Rudin [princ. math analysis] and before Lee-Top. Mani.) and J.P. May - A Concise Course in Algebraic Topology (somewhere between Dugundji and Big Rudin [real&complex analysis]). For Algebra, I recommend Axler - Linear Algebra Done Right somewhere in there (maybe before Baby Rudin) and Jacobson's Basic Algebra I & II (see tables of contents for each to see where they'd fit with Cohn's texts). Commented Nov 16, 2014 at 0:19
• Regarding general topology, it looks to me as if the first four chapters of Lee's book should be sufficient, if you only need the amount of general topology most people learn. I assume you'll be reading the "Real Analysis" part of Rudin's book as well, as that's important. I would make the fifth book (Rudin) a higher priority than the fourth (Cohn). Also, you say you have calculus covered, but few people in North America learn rigorous calculus to the right level, and Rudin only makes up for this partly. I recommend you have a glance at the table of contents o both volumes of Zorich's book...
– Mike
Commented Nov 16, 2014 at 17:44
• ... after reading Rudin's to check if there are any important topics you're missing. Some of these will be covered by Lee's Smooth Manifolds anyway, so look for other things.
– Mike
Commented Nov 16, 2014 at 17:46
• I don't know. I don't think you'll be gaining much by reading Axler's book in addition to Cohn's. I believe if you understand what's in Cohn, you won't need it, as Cohn appears to cover the same material at a higher level. I'm not sure what your goals are, but for most people it would make sense to focus on one analysis book and one algebra book first, and then re-evaluate what you want to do afterwards. Cohn is fine for algebra and either Zorich I and II, or Rudin, or Apostol's book (or another) would be fine for analysis. Really taking the time to absorb this basic analysis and algebra...
– Mike
Commented Nov 21, 2014 at 3:31
• ...is the most important thing to do, because although there is more material later on, those two subjects are where you're really introduced to the main ways of thinking in mathematics. Personally, I wouldn't want to feel as if I had to rush through this material to keep to a schedule. I'd rather take the time to enjoy it.
– Mike
Commented Nov 21, 2014 at 3:34

Assuming you're pretty well starting from scratch (that is, after taking standard first and second year courses in discrete mathematics, calculus, and linear algebra), to get a well rounded foundation in undergraduate mathematics you might consider:

• Set Theory: Halmos Naive Set Theory.
• Topology: Munkres Topology.
• Algebra: Herstein Topics in Algebra.
• Analysis: Rudin Principles.
• Geometry: do Carmo Differential Geometry of Curves and Surfaces.
• Number Theory: Burton Elementary Number Theory.

The list for 1st-year graduate level maths would be different. For this case I have written a comment below, but also consider the following links:

• Very nice list, I will have a look at these books over the next few days. Thank you! Also, do you already have a constructed list for first year graduate?
– user142198
Commented Nov 16, 2014 at 2:20
• For 1st-yr graduate I would recommend: Algebraic topology: Hatcher/Lee/Massey. Real analysis: Folland/Royden/Rudin. Complex analysis: Rudin/Ahlfors/Conway. Functional analysis: Rudin/Conway/Yosida. Algebra: Dummit&Foote/Artin/Lang. Differential geometry: Lee/Spivak/do Carmo. Number theory: Serre/Hardy&Wright/Ireland&Rosen. Algebraic geometry: Dummit&Foote/Hartshorne/Harris. Which individual book(s) you choose in each area depends on one's tastes and which book one 'takes to'.
– mdg
Commented Nov 16, 2014 at 9:29
• Thank you G.S. this is very helpful!
– user142198
Commented Nov 16, 2014 at 21:49