Topology

Cover......Page 1
Title page......Page 2
TABLE OF CONTENTS......Page 4
Errata and Addenda......Page 13
Notation and Preliminary Definitions......Page 18
Vector Spaces......Page 20
Scalar Products and Hilbert Spaces......Page 21
Normed Vector Spaces......Page 22
Metric Spaces......Page 23
Ths Space $Y^X$......Page 26
Topological Spaces (Definition)......Page 27
Closed Sets......Page 28
Induced Topology......Page 29
Homeomorphism......Page 30
Axioms of Countability......Page 31
Convergence......Page 32
Baire Category Theorem......Page 34
Compactness......Page 37
Fundamental Cube in (Separable) Hilbert Space......Page 41
Connectedness......Page 42
Topological Product......Page 43
Metrization Theorems......Page 44
Closed Convex Hull of a Set......Page 45
Points in General Position......Page 46
Definitions of Faces of a Simplex and Properly Situated Simplices......Page 49
The Canonical $n$-Simplex......Page 50
Dimension of a Complex......Page 51
Star of a Vertex......Page 52
Abstract Complex......Page 53
Geometric Realization of an $n$-Dimensional Abstract Complex in $R_{2n+1}$......Page 54
Subdivision......Page 55
Integral Oriented $r$-Chains......Page 56
Boundary Operator $\partial$......Page 57
Poincare Relation, $\partial^2 x=0$......Page 58
Scalar Product of Oriented Simplices and Chains......Page 59
Coboundaries, Cohomologous Chains and Cocycles......Page 60
Incidence and Coincidence Matrices......Page 61
Chains Over Arbitrary Abeleare Groups......Page 62
$\partial^\ast$ a Co chain Mapping......Page 63
Homomorphisms $f$, of Chains, Induced by Simplicial Mappings......Page 64
$ \partial f = f \partial $......Page 65
$ \partial^\ast f^\ast = f^\ast \partial^\ast $......Page 66
Barycenters......Page 67
Barycentric Subdivision of an Abstract Complex......Page 69
Me3h of a Geometric Complex......Page 70
$ \partial ax = x - a \partial z $......Page 71
Subdivision of Chains......Page 72
$ \partial \sigma^m = \sigma^m \partial $, $\sigma^\ast \partal^{m\ast} = \sigma^{m\ast} \partal^\ast $......Page 73
Sperner's Lemma, $\tau\sigma^m x = x $......Page 74
Definition 1 of Topological Dimension......Page 76
Monotonicity of Dimension......Page 77
Definition 2 of Topological Dimension......Page 78
Nerve and Order of a Covering......Page 80
Stars......Page 81
Top. Dim. $|K|$ = Alg. Dim. $K$......Page 82
$\varepsilon$-Mappings......Page 85
Partition of Unity Subordinate to a Covering......Page 87
Dimension Theorem for Compact Subsets of $R_n$......Page 90
The Space $I^X$......Page 91
Imbedding Theorem for Finite Dimensional Compact Metric Spaces......Page 94
Urysohn-Menger Definition of Dimension (Inductive)......Page 95
Top. Dim. $R_n = n$......Page 96
Principle of Contracting Mappings......Page 98
Lebesque Number of a Collection of Closed Sets in a Compact Space......Page 100
Covering Theorem for an $n$-Simplex......Page 101
Brouwer Fixed Point Theorem......Page 103
Theorem on the Invariance of Domain......Page 104
Application of the Theorem on Invariance of Domain......Page 108
Approximation of a Convex Compact Subset of a Banach Space by Finite Dimensional Spaces......Page 110
Schauder Fixed Point Theorem (Weak Form)......Page 111
Mazur's Lemma......Page 112
Holder Continuity......Page 114
Equicontinuous Functions and the Theorem of Arzela (Ascoli)......Page 115
Applications of Fixed Point Theorems......Page 116
Groups......Page 120
Natural Homomorphism of $A$ onto $A/B$......Page 121
Maximal Linearly Independent Set......Page 122
$\Rank A = \Rank A/B + \Rank B$......Page 124
Fundamental Theorem of Abelain Groups......Page 126
Betti Numbers and Torsion Coefficients......Page 127
Notation......Page 128
Euler-Poincare Formula......Page 130
Geometric Interpretation of the Betti Number $p_o$ of $H_o$......Page 131
Algebraie and Topological Connectedness......Page 132
Arcwise Connectedness......Page 133
Homology Groups of the Torus......Page 135
Homology Groups cf the Mobius Strip......Page 138
Homology Groups of the Projective Plane......Page 140
Pseudomanifolds......Page 142
Coherently Oriented Pseudomanifolds......Page 143
Allowable Homomorphlsm......Page 145
Theorem on a Homomorphism Between $H_r(K)$ and $H_r(L)$......Page 146
Homology Groups of a Cone......Page 147
$H_r(K) \cong H_r(\sigma K)$......Page 149
Comblnatorlally Close Slmplicial Mappings......Page 151
$\mathcal{C} K$, the Cylinder Over the Complex $K$......Page 152
Comblnatorlally Homotoplc Slmplicial Mappings......Page 156
Homomorphisms of $H^r(L)$ and $H^r(K)$......Page 158
Cohomology Groups of a $O$-Complex and a Cone......Page 159
Ordered Chains......Page 161
Ordered Homology, and Cohomology Groups......Page 162
Isomorphism of the Oriented and Ordered Homology Groups......Page 163
Isomorphism of the Oriented and Ordered Cohomology Groups......Page 166
Exterior Multiplication (Cup Product)......Page 167
$C^r$ as a Ring......Page 168
Quotient Rings......Page 169
$H^r$ as a Ring......Page 170
Differential Forms of Degree $r$......Page 171
Grassman Algebra over a Vector Space......Page 173
Graded Group with a Differential Operator, $d$......Page 174
Graded Ring with a Differential Operator......Page 175
Derived Group......Page 178
Allowable Homomorphism......Page 179
Simplicial Approximation......Page 183
Simplicial Approximation to the Identity (Sperner Mapping)......Page 185
Existence of a Simplicial Approximation to a Continuous Mapping......Page 186
Uniqueness of the Homomorphism of the Homology Groups Induced by Different Simplicial Approximations......Page 187
Homotopic Mappings......Page 188
Homotopic Mappings Induce Same Homomorphism of the Homology (Cohomology) Groups......Page 189
Homology and Cohomology Groups of a Polyhedron $X$......Page 190
Homotopic Curves......Page 192
$F(X,x) \cong F(X,y)$ if $x$ and $y$ Can be Connected by a Curve in $X$......Page 195
Product of $n$-Curves......Page 197
Commutativity of $\pi_n$ for $n > 1$......Page 198
Product of a $1$-Curve and an $n$-Curve......Page 199
Commutator Subgroup......Page 202
Paths......Page 203
Mapping, $\phi$, Qf Special Paths into Chains......Page 204
Homotopic Curves are Mapped (by $\phi$) into Homologous Chains......Page 205
For $K$ Connected, $ H_r(K,G_o)$ \cong \pi_r(|K|) $......Page 208
Homology Groups of the $n$-Sphere......Page 209
Degree of a Mapping......Page 210
Construction of a Mapping of Arbitrary Degree......Page 212
Vector Fields of Exterior and Interior Normals to $S^n$......Page 214
Hopf's Theorem: $S(f) = S(g)$ implies $f \simeq g $......Page 215
Directed Sets......Page 219
Inverse System of Groups......Page 220
Projection Mappings......Page 221
$H^r$ and $H_r$ as Direct and Inverse Systems......Page 222
Strings......Page 226
$H_r(X) = \lim_\leftarrow H_r(N(\alpha))$......Page 227
Cofinal Sets......Page 230
Isomorphism of the Cech and the Simplicial Groups for $ X = |K| $......Page 231
Alexander Cohomology Group......Page 234
Equivalence of the Cech and Alexander Groups for $X$ Compact......Page 235
Urysohn's Lemma......Page 236
Tietze Extension Theorem......Page 238
Degree of a Mapping (on $S^n$)......Page 241
Properties of Degree......Page 242
Rouche's Theorem......Page 243
Admissible Mapping......Page 246
Degree of a Mapping (in $R_n$)......Page 248
Strong Form of Homotopy Invariance and Rouche's Theorem......Page 249
Jordan-Brouwer Theorem......Page 250
Invariance of Domain......Page 252
Index......Page 254
Applications to Function Theory......Page 257
Lemmas on Mappings in $R_n$......Page 262
Completely Continuous Mappings......Page 266
Extentions of Continuous Mappings......Page 267
Allowable Approximations......Page 269
Invariance of Domain......Page 271