Modern Analysis and TopologyThe purpose of this book is to provide an integrated development of modern analysis and topology through the integrating vehicle of uniform spaces. It is intended that the material be accessible to a reader of modest background. An advanced calculus course and an introductory topology course should be adequate. But it is also intended that this book be able to take the reader from that state to the frontiers of modern analysis and topology in-so-far as they can be done within the framework of uniform spaces. Modern analysis is usually developed in the setting of metric spaces although a great deal of harmonic analysis is done on topological groups and much offimctional analysis is done on various topological algebraic structures. All of these spaces are special cases of uniform spaces. Modern topology often involves spaces that are more general than uniform spaces, but the uniform spaces provide a setting general enough to investigate many of the most important ideas in modern topology, including the theories of Stone-Cech compactification, Hewitt Real-compactification and Tamano-Morita Para compactification, together with the theory of rings of continuous functions, while at the same time retaining a structure rich enough to support modern analysis. |
Contents
III | 1 |
IV | 6 |
V | 13 |
VI | 20 |
VII | 25 |
VIII | 28 |
IX | 38 |
X | 43 |
XXXVI | 202 |
XXXVII | 203 |
XXXVIII | 210 |
XXXIX | 217 |
XL | 221 |
XLI | 229 |
XLII | 230 |
XLIII | 235 |
XI | 48 |
XII | 52 |
XIII | 56 |
XIV | 62 |
XV | 63 |
XVI | 75 |
XVII | 83 |
XVIII | 84 |
XIX | 92 |
XX | 97 |
XXI | 103 |
XXII | 110 |
XXIII | 111 |
XXIV | 114 |
XXV | 119 |
XXVI | 126 |
XXVII | 133 |
XXVIII | 146 |
XXIX | 156 |
XXX | 159 |
XXXI | 171 |
XXXII | 178 |
XXXIII | 182 |
XXXIV | 192 |
XXXV | 197 |
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Common terms and phrases
assume Baire belong Borel bounded cardinal Cauchy net CI(A CK(X Clearly closed sets clusters cofinally cofinally Cauchy cofinally complete collection compact sets compactification complete with respect complex measure Consequently containing converges COROLLARY countable defined definition denote dense disjoint equivalent exists finite open Haar measure Hausdorff space Hence homeomorphic implies inverse limit isogeneous uniform spaces isomorphism Lebesgue Lemma Lindelöf linear functional locally compact locally finite Loomis mapping measurable function metric space morphism neighborhood normal covering normal sequence o-ring open covering open set pair paracompact positive integer preparacompact Proof Proposition prove pseudo-metric quotient realcompact regular Star star-finite subcovering subspace supercomplete suppose Theorem topological group topological space topology transfinite sequence Tychonoff space U₁ uniform covering uniform space unique V₁