A Taste of Topology

This book is an introductory text for a study of topology. In general, the book is oriented to second-year undergraduates. It presents the basic language used in various fields of modern mathematics. The book covers classical topics of topology. Chapter 1 is about set theory. Chapter 2 covers metric spaces including continuity, compactness and completeness. These notions are studied in chapter 3 from the point of view of topological spaces. Separation properties are included as well. Chapter 4 is devoted to systems of continuous functions, covering the Urysohn lemma, the Stone-Čech compactification and the Stone-Weierstrass theorems. Chapter 5 presents basics of algebraic topology, containing a study of homotopy, the fundamental group and covering spaces. The text also presents several non-typical approaches to various topics. For instance, the Baire theorem is derived from Bourbaki’s Mittag-Leffler theorem, the Tychonoff theorem is proved intuitively using nets, and the complex Stone-Weierstrass theorem is obtained using a short and elegant approach. The book is also a source of exercises on basic topological notions

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