Introduction to Topology: Second Edition (Dover Books on Mathematics)

This text explains nontrivial applications of metric space topology to analysis. It covers metric space, point-set topology, and algebraic topology, and includes exercises, selected answers, and 51 illustrations. 1983 edition.

One of the most important milestones in mathematics in the twentieth century was the development of topology as an independent field of study and the subsequent systematic application of topological ideas to other fields of mathematics. While there are many other works on introductory topology, this volume employs a methodology somewhat different from other texts. Metric space and point-set topology material is treated in the first two chapters; algebraic topological material in the remaining two.

The authors lead readers through a number of nontrivial applications of metric space topology to analysis, clearly establishing the relevance of topology to analysis. Second, the treatment of topics from elementary algebraic topology concentrates on results with concrete geometric meaning and presents relatively little algebraic formalism; at the same time, this treatment provides proof of some highly nontrivial results. By presenting homotopy theory without considering homology theory, significant applications become immediately evident without the necessity of a large formal program.
Prerequisites are familiarity with real numbers and some basic set theory. Carefully chosen exercises are integrated into the text (the authors have provided solutions to selected exercises for the Dover edition). A list of notations and bibliographical references appear at the end of the book.
Dover is widely recognized for a magnificent mathematics list featuring such world-class theorists as Paul J. Cohen ( Set Theory and the Continuum Hypothesis ), Alfred Tarski ( Undecidable Theories ), Gary Chartrand ( Introductory Graph Theory ), Hermann Weyl ( The   Concept of a Riemann Surface ), Shlomo Sternberg ( Dynamical Systems ), and multiple works by C. R. Wylie in geometry, plus Stanley J. Farlow's  Partial Differential Equations for Scientists and Engineers.