Topology and Analysis

My first exposure to abstract mathematics was in an undergraduate course in topology. The textbook we used was Topology by James Munkres. I did well in the class and perceived my understanding to be rather high. However, I have been going back through the book to solidify my knowledge of the material and I have come to realize that this book is actually very in-depth and that in my course we did not fully cover what I am now finding in my second study.

My real analysis course at CU was an undergraduate course that was very much lacking in my mind and I have thus set the goal of teaching myself in a rigorous manner the entirety of analysis that is necessary for higher level mathematics. But, the question then arose how best to accomplish this? It seems that there is somewhat of a divide among mathematics over the best book for studying analysis. Royden vs Rudin. I have chosen to use Real Analysis by H.L. Royden and Real Analysis: Measure Theory, Integration, and Hilbert Spaces (Princeton Lectures in Analysis) by Elias M. Stein and Rami Shakarchi. The latter book I chose after reading the first couple chapters and finding that I rather enjoy the exposition of these authors over Royden.

I am just curious how this will turn out relative to what knowledge I might garner from retaking analysis at the graduate level or perhaps working with someone so that I concentrate on the important parts and get help where I may get stuck. Either way I think that I learn best by working on my own and this self-study should ideally also help prepare me for the independent nature of doctoral research.

The real question that keeps cropping up in my mind is whether expertise in analysis or in topology is more beneficial. It seems that analysis is more useful to the applied mathematican, of which I count myself. Hence, why study topology? But, in my reading of books on analysis and other areas I keep finding myself returning to Munkres for a topological point of view. The proofs generally seem more intuitive, which helps my understanding of the more analytic proofs. At the same time, however, much of the study of topology seems to be very abstract and hard to relate to some of the areas of applied mathematics that interest me.

In any case, I have set up some goals to work my way through much of my topology book as well as through the entirety of both analysis books. In so doing, I think that I will definently be adequetely prepared to start working on much higher levels of mathematics with a more substantial degree of confidence in my abilities.