Written by an avid user of topological groups, this book provides a concrete introduction to metric and topological groups. Important metric spaces are first introduced with the construction of a concrete metric, and a topological characterization of their open sets is immediately established. Many proofs are presented in a purely topological fashion. It is this use of topologies that affords the opportunity to discuss the limitations of some of the metric group results as well as the topological machinery needed to generalize them. The book focuses on metric cases, thus keeping many proofs in familiar territory for readers new to the topic. Readers learn how to replace the sequences with nets to obtain a general proof, and the result is an increased emphasis on topological groups and less on general topology. The exercises have been designed to help hold the interest of advanced and beginning readers, and a variety of calculations, remarks, and necessary related facts have been placed within the exercises and are used throughout the text. When used in proofs, the calculations, remarks, and facts are carefully referred to so that advanced readers will know what is needed in the proof and will be able to move forward at a faster pace. Novice readers will have to reference the exercise and workout a solution to become more engaged with the material.

Nelson G. Markley, PhD, was professor of mathematics at the University of Maryland for more than twenty-five years and also served as provost at Lehigh University. He has written numerous journal articles in the area of dynamical systems and is the author of Principles of Differential Equations, also published by Wiley.