[Books_ Math_] I have been looking for a self-study book for Analysis for a while. All the books I have looked at more or less cover the same content. They all define "compact sets" for example. However I hadn't found one that gives any motivation for why such a weird thing should be defined. Mathematicians typically do not dream up unintuitive definitions just to annoy future students. There is a problem and a historical context that makes the definition necessary. I am sure these things are intuitively obvious to the authors but for those of us not lucky enough to have had a math undergrad degree, the lack of necessary background information in textbooks is a problem. I like "The Way of Analysis" because it is the only book I have seen that tries to motivate all its definitions. Best to illustrate with some examples:
On "Open Sets and Closed Sets" (pp. 86):
"It may seem a trifling matter whether or not the endpoints are included in the interval, but it makes a significant difference for certain questions. In the open interval, every point is surrounded by a sea of other points. This is the qualitative feature we will want when we define the notion of open sets; it is certainly not true of the endpoints of the closed interval. On the other hand, an open interval (a, b) seems to be "missing" its endpoints. Although they are not points in the interval, they can be approached arbitrarily closely from within the interval. It is as if they had been unfairly omitted. The closed interval has all the points it should from this point of view, and this is the closed aspect that we will generalize when we define a closed set."
On "Compact Sets" (pp. 99)
"Infinite sets are more difficult to deal with than finite sets because of the large number of points they contain. Nevertheless, there is a class of infinite sets, called compact sets, that behave in certain limited ways very much like finite sets... In what way can an infinite set behave like a finite set? Consider the infinite pigeon-hole principle: if an infinite number of letters arrive addressed to a finite number of people, then at least one person must receive an infinite number of letters. In more conventional terms, if x1, x2, ... is an infinite sequence of real numbers, and each xj belongs to a finite set A then at least one element of A must be equal to xj for an infinite number of j's. (my note: the sequence contains only a finite number of distinct numbers, so at least one of these numbers must repeat infinite times). Now, if A were an infinite set, this statement is obviously false. However, we could hope for a slightly weaker conclusion: that A contains a limit-point of the sequence. (my note: for any infinite sequence of real numbers x1, x2, ... made up of elements from A, at least one element of A must be arbitrarily close to an infinite number of xj's) ... Let us take this property as the definition of compactness."
Compare this with the usual definition given for compactness: "A topological space T is said to be compact if every open cover of T has a finite subcover." (from Kolmogorov's Introductory Real Analysis).
Note: In Prime Obsession John Derbyshire defines Analysis simply as "the study of limits". I am ashamed to admit that this was news to me after winning a bronze medal at IMO and spending 12 years at MIT. Thus Prime Obsession is also highly recommended if you need some light reading which won't strain your brain and give you some insight into how mathematicians think.