Introduction to Metric and Topological Spaces

Introduction to Metric and Topological Spaces by Wilson Sutherland is a classic text that introduces the concepts of metric and topological spaces by illustrating their influence on continuity and convergence in mathematics. The book aims to gradually move from familiar real analysis to abstract topological spaces, using metric spaces as an intermediary. It establishes the language of metric and topological spaces with a focus on continuity, introducing several concepts first in metric spaces and then repeating them for topological spaces to build familiarity. The discussion covers connectedness, compactness, and completeness, which are widely used in various branches of mathematics. Additionally, the book explores the geometric aspect of topology, often referred to as 'rubber-sheet geometry,' with illustrations of Möbius bands, doughnuts, Klein bottles, and other surfaces, showing how this fits into the broader story of topological developments. The book is primarily aimed at second- or third-year mathematics students and includes numerous exercises, many with hints, as well as a companion website with supplementary material.