Is this study plan sufficiently general, or overly specialized?

Question

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?

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• 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

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

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List constructed with advice from comments and the one answer(added are bold). It seems to be potentially excessively long(added bold):

• 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

Answer 1

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:

• http://www.math.harvard.edu/quals/index.html
• http://math.berkeley.edu/programs/graduate/prelim-exams
• http://www.math.cornell.edu/m/Graduate/prog_content.html
• http://www.math.columbia.edu/programs-math/graduate-program/what-graduate-students-are-assumed-to-know/