Topology: An Introduction with Application to Topological Groups (Dover Books on Mathematics)

"Admirably meets the topology requirements for the pregraduate training of research mathematicians."--American Mathematical Monthly
Crucial to modern mathematics, topology is equally essential to many other disciplines, from quantum mechanics to sociology. This stimulating introduction employs the language of point set topology to define and discuss topological groups.
The text examines set-theoretic topology and its applications in function spaces, as well as homotopy and the fundamental group. This new theoretical knowledge is applied to concrete problems, such as the calculation of the fundamental group of the circle and a proof of the fundamental theorem of algebra. The abstract development concludes with the classification of topological groups by equivalence under local isomorphism.
Throughout this text, a sustained geometric development functions as a single thread of reasoning that unifies the topological course. Well-chosen exercises, along with a selection of problems in each chapter that contain interesting applications and further theory, help solidify students' working knowledge of topology and its applications.