From Wikipedia, the free encyclopedia. For the concept in graph theory, see Neighbourhood graph theory. General Topology. Undergraduate Texts in Mathematics. Translated by Sterling K. According to this definition, an open neighborhood of x is nothing more than an open subset of E that contains x.

Categories : General topology Mathematical analysis.

Others are that Euclidean n -space is homeomorphic to Euclidean -space iff , and that real valued functions achieve maxima and minima on compact sets. Foundational point-set topological questions are ones like "when can a topology on a space be derived from a metric? Point-set topology is also the ground-level of inquiry into the geometrical properties of spaces and continuous functions between them, and in that sense, it is the foundation on which the remainder of topology algebraic , differential , and low-dimensional stands.

Bing, R. Monthly 67 , Sutherland, W. New York: Oxford University Press, Vaidyanathaswamy, R. If the book needs to be three times as long, at least all that extra explanation could go into more details on the difficult topics rather than spending time on the easy points.

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## Math Doctor Bob

Another book that tries to be an easier version of papa Rudin, this is the textbook my graduate analysis class used. This one has the advantage of being much slimmer than Royden, but I found the exposition convoluted, and like Royden, Folland tends to spend time talking about the easy points and then waving his hands on the crucial, difficult points. I spent the first month of my real analysis class trying to follow along in Folland before I switched over to Rudin and never looked back.

Anyway, you can get a copy here. Advanced Roman, Advanced Linear Algebra. This is my choice for best advanced linear algebra textbook. The other books on this list are closer to reference books rather than books you can learn linear algebra from. The second part of the book is a sequence of topics in linear algebra including chapters on multilinear algebra, Hilbert spaces, applications in affine geometry, and ending with a chapter on the Umbral Calculus.

The most recent edition is the third which you can get here. One of my professors recommended this very highly, and I do recall looking here for some material on normed linear vector spaces and on Banach spaces that I was having trouble with in Roman above. Anyway, worth a look if your library has it, or if you can find a used copy on Amazon. This is the second volume in a three volume set of books on Abstract Algebra by Jacobson based on his lectures.

Ultimately, these three volumes were substantially reworked and became Basic Algebra I and Basic Algebra II reviewed in the best abstract algebra books section below. However, this second volume has a much expanded coverage of just linear algebra and is a nice standalone reference on the subject from the advanced algebraic viewpoint.

Greub, Linear Algebra , and Multilinear Algebra.

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These are the standard references at the advanced level, and deservedly so. My current university uses this textbook in all of its introductory differential equations classes so I see questions from it in the math help lab on a regular basis. Fantastic news! This book is finally going to be back in print. CRC Press has indicated they are printing a new, third edition, which they claim will be available in Hopefully, that will change with the new edition.

This book is ideal for math majors or for applied math students and engineers. This is the classic book on elementary number theory, now in its sixth edition. Every number theorist I know has a copy of this book. The prerequisites are actually very low: some exposure to calculus is probably helpful, but not strictly required.

The book is rather tightly wound and takes some work to read. Although I highly recommend this book, it really takes some dedication to get anywhere, so probably best for avowed number theorists. A nice feature is that the chapters tend to be fairly self-contained, so you can easily look up and read the chapters on continued fractions without having to worry about the material in the previous chapters. Apostol, Introduction to Analytic Number Theory. Hey, another Tom Apostol book. Like the others this one is rigorous and comprehensive and contains a lot of nice exercises.

Presumably this requires some amount of complex analysis as background, but I read it before I knew much complex analysis and still got a lot out of it. This is a nicer primer on arithmetic functions and the Riemann Zeta function. If you want a gentle introduction to the classical methods of analytic number theory and the build up to the Riemann Hypothesis, this is a great place to start. If you are self studying, I think there are solutions to most all? Silverman, Rational Points on Elliptic Curves. The main topics of the book are the Nagell-Lutz theorem and the Mordell-Weil theorem describing the points of finite order and the finite generation of the group of rational points, respectively.

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This is a new edition released in , and it added a chapter on elliptic curve cryptography. I can see this book making a good self study challenge for people in computer science that are interested in the math behind elliptic curves, or wanting to get into cryptography. A common and good choice for senior undergraduate reading or seminar courses. The first half of the book is an introduction to elementary number theory, while the second half is mostly more advanced topics in number theory requiring some abstract algebra.

I think the first half is completely accessible without much background, but the algebraic prerequisites definitely get higher as the book moves on. There are nice sections on zeta functions, L-functions, algebraic number fields, and elliptic curves. Definitely the place to go for a broad and modern introduction to number theory if you have sufficient maturity to read it or can develop sufficient maturity as you read it to keep up.

It also has a ton of really nice exercises.

## Math Doctor Bob

Samuel, Algebraic Theory of Numbers. A tiny sliver of a book translated from French. My undergraduate algebraic number theory class used this a few years ago and I had to hunt down and pay a fortune for a used copy of the old edition. Serre, A Course in Arithmetic. A lovely book on arithmetic by Serre. Apocryphal stories about people mistaking the book for a book on elementary school arithmetic abound. There are no exercises, and the book is quite terse, but unravelling what Serre is doing takes a lot of work.

This is really at the edge of what could profitably be read by undergraduates, but with some supervision there is plenty of accessible material in here. It will certainly stimulate some development in your ability to deal with tightly written books. This is an undergraduate level introduction to complex variables and complex analysis. Needham, Visual Complex Analysis. This is a very popular companion book for complex analysis students. Developing the ability to visualize what is going on in complex analysis is a big asset, as your geometric intuition is a powerful tool.

This is a really nice book for that purpose, but I actually prefer the for some reason much less popular book by Wegert in the intermediate section below. This two volume set is my top choice for learning complex analysis at the introductory graduate level. The first four chapters of Volume 1 cover the standard complex analysis material everyone is expected to know though in my experience graduate courses never seem to explicitly cover on integration and differentiation in , analytic continuation, etc.

The coverage is brisk, but thorough. The second half of Volume 2 is devoted to topics in complex analysis. In particular, there are chapters on elliptic functions and elliptic modular forms. I also think this is the best complex analysis book for self study as complete solutions to all of the exercises of Volume 1 are provided. Volume 2 probably belongs on the Advanced Topics list. It contains an introduction to Riemann surfaces, abelian functions, and modular forms of several variables.

Note: I linked to the existing printing, but Dover is reprinting this book in the near ish future. Intermediate Tu, An Introduction to Manifolds. I took a graduate course in differential geometry two years ago and would have failed without this book. Loring Tu wrote this to cover the material a student should know before starting his famed book with Raoul Bott, Differential Forms in Algebraic Topology.

This is the most lucid introduction to manifolds I know of, and conveys the material with remarkable clarity. Milnor, Topology from the Differentiable Viewpoint. Approximately every book Milnor wrote is a classic. One of the truly great mathematical writers and one of the truly great mathematicians. The book is only 80 pages, and would be a great supplement in any geometry course. The other book that got me through my graduate differential geometry class. This is exactly what the title says, a workbook of smooth manifold problems. There are problems total, all with complete, detailed solutions really detailed.

The differential geometry book for physicists and students in applied math. A very different flavor from the pure mathematics books elsewhere on the list.

### Freely available

The physics majors I know love this book. Lee, An Introduction to Smooth Manifolds. Ah, Lee. A monster of a book, packed with information. This was the official textbook for my grad differential geometry class and I really struggled with it my first time through.

I wish there were more worked out examples in the text.

I recommend the books by Tu and Gadea in the Intermediate section above as supplements. So, if you want to bang your head against a wall and maybe learn some differential geometry in the process, look no further. Also, be sure to get the second edition as there were pretty big changes from the first.

This is a classic that everyone recommends and some people truly love.