General Topology, Chapters 1-4 (Elements of Mathematics) 〈1〉

General Topology, Chapters 1-4 (Elements of Mathematics) 〈1〉

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Table of Contents

Advice to the Reader                               v
Contents of the Elements of Mathematics Series 9 (2)
Introduction 11 (6)
Topological Structures 17 (152)
Open sets, neighbourhoods, closed sets 17 (8)
Open sets 17 (1)
Neighbourhoods 18 (3)
Fundamental Systems of neighbourhoods; 21 (1)
bases of a topology
Closed sets 21 (1)
Locally finite families 22 (1)
Interior, closure, frontier of a set; 23 (2)
dense sets
Continuous functions 25 (10)
Continuous functions 25 (3)
Comparison of topologies 28 (2)
Initial topologies 30 (2)
Final topologies 32 (2)
Pasting together of topological spaces 34 (1)
Subspaces, quotient spaces 35 (8)
Subspaces of a topological space 35 (2)
Continuity with respect to a subspace 37 (1)
Locally closed subspaces 38 (1)
Quotient spaces 39 (1)
Canonical decomposition of a continuous 40 (2)
mapping
Quotient space of a subspace 42 (1)
Product of topological spaces 43 (7)
Product spaces 43 (3)
Section of an open set; section of a 46 (1)
closed set, projection of an open set.
Partial continuity
Closure in a product 47 (1)
Inverse limits of topological spaces 48 (2)
Open mappings and closed mappings 50 (7)
Open mappings and closed mappings 50 (2)
Open equivalence relations and closed 52 (2)
equivalence relations
Properties peculiar to open mappings 54 (2)
Properties peculiar to closed mappings 56 (1)
Filters 57 (11)
Definition of a filter 57 (1)
Comparison of filters 58 (1)
Bases of a filter 59 (1)
Ultrafilters 60 (1)
Induced filter 61 (1)
Direct image and inverse image of a 62 (1)
filter base
Product of filters 63 (1)
Elementary filters 64 (1)
Germs with respect to a filter 65 (3)
Germs at a point 68 (1)
Limits 68 (7)
Limit of a filter 68 (1)
Cluster point of a filter base 69 (1)
Limit point and cluster point of a 70 (2)
function
Limits and continuity 72 (1)
Limits relative to a subspace 73 (1)
Limits in product spaces and quotient 74 (1)
spaces
Hausdorff spaces and regular spaces 75 (8)
Hausdorff spaces 75 (2)
Subspaces and products of Hausdorff spaces 77 (1)
Hausdorff quotient spaces 78 (2)
Regular spaces 80 (1)
Extension by continuity; double limit 81 (1)
Equivalence relations on a regular space 82 (1)
Compact spaces and locally compact spaces 83 (14)
Quasi-compact spaces and compact spaces 83 (2)
Regularity of a compact space 85 (1)
Quasi-compact sets; compact sets; 85 (2)
relatively compact sets
Image of a compact space under a 87 (1)
continuous mapping
Product of compact spaces 88 (1)
Inverse limits of compact spaces 89 (1)
Locally compact spaces 90 (2)
Embedding of a locally compact space in a 92 (1)
compact space
Locally compact σ-compact spaces 93 (1)
Paracompact spaces 94 (3)
Proper mappings 97 (10)
Proper mappings 97 (4)
Characterization of proper mappings by 101(3)
compactness properties
Proper mappings into locally compact 104(1)
spaces
Quotient spaces of compact spaces and 105(2)
locally compact spaces
Connectedness 107(62)
Connected spaces and connected sets 107(2)
Image of a connected set under a 109(1)
continuous mapping
Quotient spaces of a connected space 110(1)
Product of connected spaces 110(1)
Components 110(2)
Locally connected spaces 112(1)
Application: the Poincare-Volterra theorem 113(4)
Exercises for § 1 117(2)
Exercises for § 2 119(3)
Exercises for § 3 122(3)
Exercises for § 4 125(2)
Exercises for § 5 127(2)
Exercises for § 6 129(3)
Exercises for § 7 132(1)
Exercises for § 8 133(8)
Exercises for § 9 141(9)
Exercises for § 10 150(5)
Exercises for § 11 155(7)
Historical Note 162(5)
Bibliography 167(2)
Uniform Structures 169(50)
Uniform spaces 169(5)
Definition of a uniform structure 169(2)
Topology of a uniform space 171(3)
Uniformly continuous functions 174(7)
Uniformly continuous functions 174(1)
Comparison of uniformities 175(1)
Initial uniformities 176(1)
Inverse image of a uniformity; uniform 177(1)
subspaces
Least upper bound of a set of uniformities 178(1)
Product of uniform spaces 179(1)
Inverse limits of uniform spaces 180(1)
Complete spaces 181(17)
Cauchy filters 181(2)
Minimal Cauchy filters 183(1)
Complete spaces 184(2)
Subspaces of complete spaces 186(1)
Products and inverse limits of complete 186(4)
spaces
Extension of uniformly continuous 190(1)
functions
The completion of a uniform space 191(4)
The Hausdorff uniform space associated 195(2)
with a uniform space
Completion of subspaces and product spaces 197(1)
Relations between uniform spaces and 198(21)
compact spaces
Uniformity of compact spaces 198(2)
Compactness of uniform spaces 200(3)
Compact sets in a uniform space 203(1)
Connected sets in a compact space 204(2)
Exercises for § 1 206(1)
Exercises for § 2 207(1)
Exercises for § 3 208(1)
Exercises for § 4 209(7)
Historical Note 216(2)
Bibliography 218(1)
Topological Groups 219(110)
Topologies on groups 219(6)
Topological groups 219(2)
Neighbourhoods of a point in a 221(3)
topological group
Isomorphisms and local isomorphisms 224(1)
Subgroups, quotient groups, homomorphisms, 225(17)
homogeneous spaces, product groups
Subgroups of a topological group 225(2)
Components of a topological group 227(1)
Dense subgroups 228(1)
Spaces with operators 228(3)
Homogeneous spaces 231(1)
Quotient groups 232(1)
Subgroups and quotient groups of a 233(2)
quotient group
Continuous homomorphisms and strict 235(2)
morphisms
Products of topological groups 237(2)
Semi-direct products 239(3)
Uniform structures on groups 242(8)
The right and left uniformities on a 242(2)
topological group
Uniformities on subgroups, quotient 244(1)
groups and product groups
Complete groups 245(1)
Completion of a topological group 246(2)
Uniformity and completion of a 248(2)
commutative topological group
Groups operating properly on a topological 250(11)
space; compactness in topological groups
and spaces with operators
Groups operating properly on a 250(3)
topological space
Properties of groups operating properly 253(1)
Groups operating freely on a topological 254(1)
space
Locally compact groups operating properly 255(2)
Groups operating continuously on a 257(2)
locally compact space
Locally compact homogeneous spaces 259(2)
Infinite sums in commutative groups 261(10)
Summable families in a commutative group 261(1)
Cauchy's criterion 262(2)
Partial sums; associativity 264(2)
Summable families in a product of groups 266(1)
Image of a summable family under a 267(1)
continuous homomorphism
Series 267(2)
Commutatively convergent series 269(2)
Topological groups with operators; 271(13)
topological rings, division rings and fields
Topological groups with operators 271(1)
Topological direct sum of stable subgroups 272(2)
Topological rings 274(2)
Subrings; ideals; quotient rings; 276(1)
products of rings
Completion of a topological ring 276(2)
Topological modules 278(3)
Topological division rings and fields 281(1)
Uniformities on a topological division 282(2)
ring
Inverse limits of topological groups and 284(45)
rings
Inverse limits of algebraic structures 284(2)
Inverse limits of topological groups and 286(3)
spaces with operators
Approximation of topological groups 289(4)
Application to inverse limits 293(3)
Exercises for § 1 296(2)
Exercises for § 2 298(8)
Exercises for § 3 306(2)
Exercises for § 4 308(6)
Exercises for § 5 314(1)
Exercises for § 6 315(9)
Exercises for § 7 324(3)
Historical Note 327(1)
Bibliography 328(1)
Real Numbers 329(90)
Definition of real numbers 329(5)
The ordered group of rational numbers 329(1)
The rational line 330(1)
The real line and real numbers 331(1)
Properties of intervals in R 332(1)
Length of an interval 333(1)
Additive uniformity of R 334(1)
Fundamental topological properties of the 334(5)
real line
Archimedes' axiom 334(1)
Compact subsets of R 335(1)
Least upper bound of a subset of R 335(1)
Characterization of intervals 336(1)
Connected subsets of R 336(2)
Homeomorphisms of an interval onto an 338(1)
interval
The field of real numbers 339(3)
Multiplication in R 339(1)
The multiplicative group R* 340(1)
nth roots 341(1)
The extended real line 342(5)
Homeomorphism of open intervals of R 342(1)
The extended line 343(2)
Addition and multiplication in R 345(2)
Real-valued functions 347(12)
Real-valued functions 347(1)
Real-valued functions defined on a 348(1)
filtered set
Limits on the right and on the left of a 349(1)
function of a real variable
Bounds of a real-valued function 350(2)
Envelopes of a family of real-valued 352(1)
functions
Upper limit and lower limit of a 353(3)
real-valued function with respect to a
filter
Algebraic operations on real-valued 356(3)
functions
Continuous and semi-continuous real-valued 359(4)
functions
Continuous real-valued functions 359(1)
Semi-continuous functions 360(3)
Infinite sums and products of real numbers 363(10)
Families of positive finite numbers 364(2)
summable in R
Families of finite numbers of arbitrary 366(1)
sign summable in R
Product of two infinite sums 367(1)
Families multipliable in R* 367(2)
Summable families and multipliable 369(1)
families in R
Infinite series and infinite products of 370(3)
real numbers
Usual expansions of real numbers; the power 373(46)
of R
Approximations to a real number 373(1)
Expansions of real numbers relative to a 373(1)
base sequence
Definition of a real number by means of 374(2)
its expansion
Comparison of expansions 376(1)
Expansions to base a 376(1)
The power of R 377(1)
Exercises for § 1 378(3)
Exercises for § 2 381(6)
Exercises for § 3 387(1)
Exercises for § 4 388(1)
Exercises for § 5 389(4)
Exercises for § 6 393(4)
Exercises for § 7 397(4)
Exercises for § 8 401(5)
Historical Note 406(11)
Bibliography 417(2)
Index of Notation (Chapters I-IV) 419(2)
Index of Terminology (Chapters I-IV) 421