This book provides an elementary introduction to the ideas and methods of topology by the detailed study of certain topics. There are elegant but rigorous proofs of many of the basic theorems, including Jordan's curve theorem and theorem inverse to Jordan's theorem.
Author(s): M. H. A. Newman
Edition: reprinted 2-nd edition
Year: 1964
Language: English
Pages: 214
Chapter 1. SETS
§ 1. The calculus of sets
§ 2. Enumerable and non-enumerable sets
Chapter II. CLOSED SETS AND OPEN SETS IN METRIC SPACES
§ 1. Closed and open sets
§ 2. Convergent sequences of points. Compact sets
§ 3. Induced structure. Relative and absolute properties
§ 4. Complete spaces
Chapter III. HOMEOMORPHISM AND CONTINUOUS MAPPINGS
§ 1. Equivalent metrics. Topological spaces. Homeomorphism
§ 2. Continuous mappings
Chapter IV. CONNECTION
§ 1. Connected sets
§ 2. Components
§ 3. Connection in compact spaces
§ 4. Local connection
§ 5. Topological characterisation of segment and circle
Chapter V. SEPARATION THEOREMS
§ 1. Chains on a grating
§ 2. Alexander’s Lemma and Jordan’s Theorem
§ 3. Invariance of dimension number and of open sets
§ 4. Further separation theorems
§ 5. Extension to sets of points in Z^p and R^p
Chapter VI. SIMPLY- AND MULTIPLY-CONNECTED PLANE
DOMAINS
§ 1. Simply-connected domains
§ 2. Mapping on a standard domain
§ 3. Connectivity of open sets
§ 4. Relations of a domain to its frontier
§ 5. Topological mapping of Jordan domains and their
closures
Chapter VII. HOMOTOPY PROPERTIES
§ 1. Paths and deformations
§ 2. Intersection and orientation of paths in R^2
