NICOLAS BOURBAKI ELEMENTS OF MATHEMATICS. General Topology. Chapters 1-4. Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

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1 NICOLAS BOURBAKI ELEMENTS OF MATHEMATICS General Topology Chapters 1-4 Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

2 ADVICE TO THE READER v CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES 9 INTRODUCTION 11 CHAPTER I. Topological Structures Open sets, neighbourhoods, closed sets Open sets Neighbourhoods Fundamental systems of neighbourhoods; bases of a topology Closed sets Locally finite families Interior, closure, frontier of a set; dense sets Continuous functions Continuous functions Comparison of topologies Initial topologies Final topologies Pasting together of topological spaces Subspaces, quotient spaces Subspaces of a topological space Continuity with respect to a subspace Locally closed subspaces Quotient spaces Canonical decomposition of a continuous mapping Quotient space of a subspace 42 1

3 4. Product of topological spaces Product spaces Section of an open set; section of a closed set, projection of an open set. Partial continuity Closure in a product Inverse limits of topological spaces Open mappings and closed mappings Open mappings and closed mappings Open equivalence relations and closed equivalence relations Properties peculiar to open mappings Properties peculiar to closed mappings Filters Definition of a filter Comparison of filters Bases of a filter Ultrafilters Induced filter Direct image and inverse image of a filter base Product of filters Elementary filters Germs with respect to a filter 65 io. Germs at a point Limits Limit of a filter Cluster point of a filter base Limit point and cluster point of a function Limits and continuity Limits relative to a subspace Limits in product spaces and quotient spaces HausdorfF spaces and regular spaces HausdorfF spaces Subspaces and products of HausdorfF spaces HausdorfF quotient spaces Regular spaces Extension by continuity; double limit Equivalence relations on a regular space Compact spaces and locally compact spaces Quasi-compact spaces and compact spaces Regularity of a compact space 85 2

4 3. Quasi-compact sets; compact sets; relatively compact sets Image of a compact space under a continuous mapping Product of compact spaces Inverse limits of compact spaces Locally compact spaces Embedding of a locally compact space in a compact space Locally compact u-compact spaces Paracompact spaces Proper mappings Proper mappings Characterization of proper mappings by compactness properties Proper mappings into locally compact spaces Quotient spaces of compact spaces and locally compact spaces Connectedness I07 1. Connected spaces and connected sets Image of a connected set under a continuous mapping Quotient spaces of a connected space no 4. Product of connected spaces no 5. Components no 6. Locally connected spaces Application : the Poincare-Volterra theorem 113 Exercises for Exercises for Exercises for Exercises for Exercises for Exercises for Exercises for Exercises for Exercises for Exercises for Exercises for n 155 Historical Note 162 Bibliography 167 3

5 CHAPTER II. Uniform Structures 169 i. Uniform spaces Definition of a uniform structure Topology of a uniform space Uniformly continuous functions Uniformly continuous functions Comparison of uniformities Initial uniformities Inverse image of a uniformity; uniform subspaces Least upper bound of a set of uniformities Product of uniform spaces Inverse limits of uniform spaces Complete spaces Cauchy niters Minimal Cauchy niters Complete spaces Subspaces of complete spaces Products and inverse limits of complete spaces Extension of uniformly continuous functions The completion of a uniform space The Hausdorff uniform space associated with a uniform space Completion of subspaces and product spaces Relations between uniform spaces and compact spaces Uniformity of compact spaces Compactness of uniform spaces Compact sets in a uniform space Connected sets in a compact space 204 Exercises for Exercises for Exercises for Exercises for Historical Note 216 Bibliography 218 CHAPTER III : Topological Groups Topologies on groups Topological groups 219

6 2. Neighbourhoods of a point in a topological group Isomorphisms and local isomorphisms Subgroups, quotient groups, homomorphisms, homogeneous spaces, product groups Subgroups of a topological group Components of a topological group Dense subgroups Spaces with operators Homogeneous spaces Quotient groups Subgroups and quotient groups of a quotient group Continuous homomorphisms and strict morphisms Products of topological groups Semi-direct products Uniform structures on groups The right and left uniformities on a topological group Uniformities on subgroups, quotient groups and product groups Complete groups Completion of a topological group Uniformity and completion of a commutative topological group Groups operating properly on a topological space; compactness in topological groups and spaces with operators Groups operating properly on a topological space Properties of groups operating properly Groups operating freely on a topological space Locally compact groups operating properly Groups operating continuously on a locally compact space Locally compact homogeneous spaces Infinite sums in commutative groups Summable families in a commutative group Cauchy's criterion Partial sums; associativity Summable families in a product of groups Image of a summable family under a continuous homomorphism Series Commutatively convergent series 269 5

7 6. Topological groups with operators; topological rings, division rings and fields Topological groups with operators Topological direct sum of stable subgroups Topological rings Subrings; ideals; quotient rings; products of rings Completion of a topological ring Topological modules Topological division rings and fields Uniformities on a topological division ring Inverse limits of topological groups and rings Inverse limits of algebraic structures Inverse limits of topological groups and spaces with operators Approximation of topological groups Application to inverse limits 293 Exercises for Exercises for Exercises for Exercises for Exercises for Exercises for Exercises for Historical Note 327 Bibliography 328 CHAPTER IV : Real Numbers Definition of real numbers The ordered group of rational numbers The rational line The real line and real numbers Properties of intervals in R Length of an interval Additive uniformity of R Fundamental topological properties of the real line Archimedes' axiom Compact subsets of R Least upper bound of a subset of R Characterization of intervals 336

8 5. Connected subsets of R Homeomorphisms of an interval onto an interval The field of real numbers Multiplication in R The multiplicative group R* nth roots The extended real line Homeomorphism of open intervals of R The extended line Addition and multiplication in R Real-valued functions Real-valued functions Real-valued functions defined on a filtered set Limits on the right and on the left of a function of a real variable Bounds of a real-valued function Envelopes of a family of real-valued functions Upper limit and lower limit of a real-valued function with respect to a filter Algebraic operations on real-valued functions Continuous and semi-continuous real-valued functions Continuous real-valued functions Semi-continuous functions Infinite sums and products of real numbers Families of positive finite numbers summable in R Families of finite numbers of arbitrary sign summable in R Product of two infinite sums Families multipliable in R* Summable families and multipliable families in R Infinite series and infinite products of real numbers Usual expansions of real numbers; the power of R Approximations to a real number Expansions of real numbers relative to a base sequence Definition of a real number by means of its expansion Comparison of expansions 376 7

9 5. Expansions to base a The power of R 377 Exercises for Exercises for Exercises for Exercises for Exercises for Exercises for Exercises for Exercises for Historical Note 406 Bibliography 417 Index of Notation (Chapters I-IV) 419 Index of Terminology (Chapters I-IV) 421 8