Real analysis and Topology book recommendetion

Question

I hold a bachelor's degree in mathematics and I have taken undergraduate real analysis course. But I'm interested to know more about it, especially the stuff that will help me to understand more of topology. Could you please recommend me some books that I can start with?

Your help will be appreciated

Thanks in advance.

Answer 1

You might enjoy an analysis book which has strong emphasis on function spaces as an application of general metric space theory. A good one along these lines is Carothers' Real Analysis. The book is divided into three parts: metric spaces, function spaces, and Lebesgue measure/integration.

A typical quote which illustrates the spirit of the book:

In the next few chapters we will focus our attentions on $C[0,1]$ (the space of continuous functions from $[0,1]$ to $\mathbb R$) and some of its relatives. We will want to answer all the same questions that we have asked of every other metric space. What are its open sets? Its compact sets? Is $C[0,1]$ complete? Is it separable? And on and on. You name it, we want to know it.

Answer 2

Real analysis : authors: Folland, Rudin, Royden (hist newest edition), Frank jones (especially for gamma functions)

Point set topology: authors: Munkres, Hocking & Young (this is not standard, but I like it personally)

In my very opinion, one should learn multivariable calculus before studying real analysis. I prefer texts written in abstract spirit and here is the best multivariable text imo : Cartan - Differential Calculus

Answer 3

The old classic General Topology by John Kelley has a strong emphasis on analysis. I prefer it to Munkres for being better structured and more concise. That said, it is still a topology book, and I am uncertain if it is really the case that every young analyst should know all it covers (as suggested by Kelley himself sixty years ago). A few do deem it to be obsolete.

Functional analysis is a subject where topology is heavily emphasized, especially when it comes to general topological vector spaces. In theory it is possible to study it without knowledge of Lebesgue integration, although many concrete examples will be lost in the process. But even if you have not learned Lebesgue integration before, it is still feasible to study both at the same time, as I once did. As for books, nothing beats Rudin for me.

Answer 4

Carothers' book that Bungo mentioned seems like it would be a good fit for you, but given what you wrote I think the following might be an even better fit:

Introduction to Topology and Modern Analysis by George F. Simmons