analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
…
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Handbook of Analysis and its Foundations
Academic Press (1996)
(errata)
on analysis.
Schechter's Handbook of Analysis and its Foundations is a large book, intended for self study by beginning graduate students or senior-level undergraduates, on all of the basic topics of abstract analysis and then some. Its logical flow is very much like that of Bourbaki, but focussed on that which applies to analysis and written in a modern style. Except for the slightly broken index (check the errata!), it is very user-friendly, with sketched proofs phrased as exercises with hints, many examples (eventually), and abundant cross references. It is also extremely self contained; the only prerequisite is mathematical maturity, and the first section even helps with that!
It begins, as the name implies, with foundations: not only the usual naïve set theory, but also a discussion of ZFC, constructive mathematics, and enough model theory to do nonstandard analysis. There is special emphasis on the axiom of choice; throughout the book, it is explicitly pointed out whenever anything beyond dependent choice and excluded middle is required. (The axioms of replacement and foundation, on the few occasions when they appear, are also pointed out, so logically the book takes place in ${Z^-} + {DC}$.)
The book then moves on to algebra, moving from monoids to fields, on the grounds that such algebra also serves as a foundation for analysis. This culminates in a treatment of category theory; this is somewhat unsatisfactory (although very good for an analysis book!) and is not much more than Bourbaki's theory of structures reinterpreted as a theory of concrete categories. After algebra comes topology, and then analysis proper.
Those aspects of analysis that do not depend on the theory of the real numbers are also covered when appropriate in the set-theory and algebra sections of the book; thus topological spaces and convex sets (for example) are defined early, although they are not thoroughly studied until later (after the foundational parts are finished). The analysis in the book is soft and abstract, on the grounds that this material serves as the proper foundation for hard results in concrete cases. However, many concrete examples are given to illustrate the abstract ideas. The breadth of topics covered, even within analysis itself, is quite wide; from convergence spaces to ultrabarrel?s, from the Henstock integral to the Brouwer fixed-point theorem, it has it all.
But everything must stop somewhere; it does not cover complex analysis.