Foundations of topology/
C. Wayne Patty.
- 2nd ed.
- Boston: Jones and Bartlett, 2013.
- xi, 380 p. : ill. ; 24 cm.
- Jones and Bartlett Publishers series in mathematics., Topology. .
1 Topological Spaces 1 1 Metric Spaces 1 2 Topological Spaces: The Definition and Examples 10 3 Basis for a Topology 15 4 Closed Sets, Closures, and Interiors of Sets 26 5 Metric Spaces Revisited 36 6 Convergence 42 7 Continuous Functions and Homeomorphisms 50 2 New Spaces from Old Ones 59 1 Subspaces 59 2 The Product Topology on X ? Y 67 3 The Product Topology 73 4 The Weak Topology and the Product Topology 80 5 The Uniform Metric 85 6 Quotient Spaces 90 3 Connectedness 102 1 Connected Spaces 102 2 Pathwise and Local Connectedness 112 3 Totally Disconnected Spaces 119 4 Compactness 123 1 Compactness in Metric Spaces 123 2 Compact Spaces 131 3 Local Compactness and the Relation Between Various Forms of Compactness 140 4 The Weak Topology on a Topological Space 145 5 Equicontinuity 149 5 The Separation and Countability Axioms 154 1 T0-, T1-, and T2-Spaces 154 2 Regular and Completely Regular Spaces 159 3 Normal and Completely Normal Spaces 167 4 The Countability Axioms 172 5 Urysohn¿s Lemma and the Tietze Extension Theorem 176 6 Embeddings 180 6 Special Topics 185 1 Contraction Mappings in Metric Spaces 185 2 Normed Linear Spaces 187 3 The Fréchet Derivative 192 4 Manifolds 199 5 Fractals 210 6 Compactifications 218 7 The Alexander Subbase and the Tychonoff Theorems 225 7 Metrizability and Paracompactness 229 1 Urysohn¿s Metrization Theorem 229 2 Paracompactness 232 3 The Nagata-Smirnov Metrization Theorem 241 8 The Fundamental Group and Covering Spaces 248 1 Homotopy of Paths 248 2 The Fundamental Group 257 3 The Fundamental Group of the Circle 261 4 Covering Spaces 265 5 Applications and Additional Examples of Fundamental Groups 268 6 Knots 274 9 Applications of Homotopy 281 1 Inessential Maps 281 2 The Fundamental Theorem of Algebra 283 3 Homotopic Maps 285 4 The Jordan Curve Theorem 287 A Logic and Proofs 294 B Sets 300 C Functions 305 D Indexing Sets and Cartesian Products 311 E Equivalence Relations and Order Relations 315 F Countable Sets 320 G Uncountable Sets 324 H Ordinal and Cardinal Numbers 326 I Algebra 331 Bibliography 338 Index 340