Topology : point-set and geometric / Paul L. Shick.

Front Matter -- Introduction: Intuitive Topology -- Background on Sets and Functions -- Topological Spaces -- More on Open and Closed Sets and Continuous Functions -- New Spaces from Old -- Connected Spaces -- Compact Spaces -- Separation Axioms -- Metric Spaces -- The Classification of Surfaces -- Fundamental Groups and Covering Spaces -- References -- Index -- Pure and Applied Mathematics.

Includes bibliographical references (pages 263-264) and index.

foreword -- Acknowledgments -- 1. Introduction : Intuitive topology -- 1.1. Introduction : intuitive topology -- 2. Background on sets and functions -- 2.1. Sets -- 2.2. Functions -- 2.3. Equivalence relations -- 2.4. Induction -- 2.5. Cardinal numbers -- 2.6. Groups -- 3. Topological spaces -- 3.1. Introduction -- 3.2. Definitions and examples -- 3.3. Basics on open and closed sets -- 3.4. The subspace topology -- 3.5. Continuous functions -- 4. More on open and closed sets and continuous functions -- 4.1. Introduction -- 4.2. Basis for a topology -- 4.3. Limit points -- 4.4. Interior, boundary and closure -- 4.5. More on continuity -- 5. New spaces from old -- 5.1. Introduction -- 5.2. Product spaces -- 5.3. Infinite product spaces (optional) -- 5.4. Quotient spaces -- 5.5. Unions and wedges -- 6. Connected spaces -- 6.1. Introduction -- 6.2. Definition, examples and properties -- 6.3. Connectedness in the real line -- 6.4. Path-connectedness -- 6.5. Connectedness of unions and finite products -- 6.6. Connnectedness of infinite products (optional) -- 7. Compact spaces -- 7.1. Introduction -- 7.2. Definition, examples and properties -- 7.3. Hausdorff spaces and compactness -- 7.4. Compactness in the real line -- 7.5. Compactness of products -- 7.6. Finite intersection property (optional).

8. Separation axioms -- 8.1. Introduction -- 8.2. Definition and examples -- 8.3. Regular and normal spaces -- 8.4. Separation axioms and compactness -- 9. Metric spaces -- 9.1. Introduction -- 9.2. Definition and examples -- 9.3. Properties of metric spaces -- 9.4. Basics on sequences -- 10. The classification of surfaces -- 10.1. Introduction -- 10.2. Surfaces and higher-dimensional manifolds -- 10.3. Connected sums of surfaces -- 10.4. The classification theorem -- 10.5. Triangulations of surfaces -- 10.6. Proof of the classification theorem -- 10.7. Euler characteristics and uniqueness -- 11. Fundamental groups and covering spaces -- 11. 1. Introduction -- 11.2. Homotopy of functions and paths -- 11.3. An operation on paths -- 11.4. The fundamental group -- 11.5. Covering spaces -- 11.6. Fundamental group of the circle and related spaces -- 11.7. The fundamental groups of surfaces -- References -- Index.

This text covers the essentials of point-set topology in a relatively terse presentation, with lots of examples and motivation along the way. Along with the standard point-set topology topics (connected spaces, compact spaces, separation axioms, and metric spaces), the author includes path-connectedness, and a chapter on constructing spaces from other spaces (including products, quotients, etc.). The text culminates in to two main chapters, each independent of the other: 1) The Classification Theorem for Compact, Connected Surfaces and 2) Fundamental Groups and Covering Spaces, with Applications giving the reader the choice of which subject best suits them.

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