Topological spaces: from distance to neighborhood

gentle introduction to the subject, leading the reader to understand the notion of what is important in topology with regard to geometry. Divided into three sections - The line and the plane, Metric spaces and Topological spaces -, the book eases the move into higher levels of abstraction. Students are thereby informally assisted in learning new ideas while remaining on familiar territory. The authors do not assume previous knowledge of axiomatic approach or set theory. Similarly, they have restricted the mathematical vocabulary in the book so as to avoid overwhelming the reader, and the concept of convergence is employed to allow students to focus on a central theme while moving to a natural understanding of the notion of topology. The pace of the book is relaxed with gradual acceleration: the first nine sections form a balanced course in metric spaces for undergraduates while also containing ample material for a two-semester graduate course. Finally, the book illustrates the many connections between topology and other subjects, such as analysis and set theory, via the inclusion of "Extras" at the end of each chapter presenting a brief foray outside topology.

Contents: Preface.- The Line and the Plane. What Topology is about; Axioms for R.; Convergent sequences and continuity; Curves in the plane.- Metric Spaces. Metrics; Open and closed sets; Completeness; Uniform convergence; Sequential compactness; Convergent nets; Transition to Topology.- Topological Spaces. Topological spaces; Compactness and the Hausdorff property; Products and quotients; The Hahn-Tietze-Tong-Urysohn Theorems; Connectedness; Tychonoff's Theorem.- Postscript. A Smorgasbord for further study; Countable sets.- A Farewell to the Reader.- Literature.- Index of Symbols.- Index of Terms.

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