Elements of Abstract Analysis
In nature's infinite book ofsecrecy A little I can read. Antony and Cleopatra, l. ii. This is a book about a few elementary concepts of analysis and the mathe matical structures which enfold them. It is more concerned with the interplay amongst these concepts than with their many applications. The book is self-contained; in the first chapter, after acknowledging the fundamental role ofmathematical logic, wepresent seven axioms of Set Theory; everything else is developed from these axioms. It would therefore be true, if misleading, to say that the reader requires no prior knowledge of mathematics. In reality, the reader we have in mind has that level of sophistication achieved in about three years of undergraduate study of mathematics and is already well acquainted with most of the structures discussed-rings, linear spaces, metric spaces, and soon-and with many ofthe principal analytical concepts convergence, connectedness, continuity,compactness and completeness. Indeed, it is only after gaining familiarity with these concepts and their applications that it is possible to appreciate their place within a broad framework of set based mathematics and to consolidate an understanding of them in such a framework. To aid in these pursuits, wepresent our reader with things familiar and things new side by side in most parts of the book-and we sometimes adopt an unusual perspective. That this is not an analysis textbook is clear from its many omissions.
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Continuity and Openness
arbitrary Axiom of Choice bijective bounded called cardinal closed subset commutative compact space converges convex Corollary counting number defined Definition denoted dense disjoint dom(r domain endowed ensures equinumerous equivalent Example EXERCISES Q exists f is continuous field F filter finite subset follows G R+ Hausdorff space Hilbert space homomorphism includes induction inequality initial topology injective injective function intersection inverse Lemma linear subspace linearly independent mathematics maximal subspace metric space nbd(x non-empty set non-empty subset non-trivial normed linear space notation open ball open intervals open neighbourhood open sets open subset order isomorphism ordered set ordinal partially ordered Proof Let Proof Suppose ps(X real numbers Recursive relative topology second countable semimetric space seminormed seminormed linear space Show subbase surjective topological space topology determined totally ordered ultrafilter union unique unit ball vector space whence