Topology is the study of when one geometric object can be bent and twisted, but not ripped, into another object. Determining when two objects are topologically the same is incredibly difficult and is still the subject of a tremendous amount of research. The first part of the course involves establishing a precise definition for topological equivalence (point-set topology). Then we develop methods to determine when objects are the same. We will define homotopy and homology and, if time permits, examine the topological classifications of surfaces, which will show that you cannot twist a basketball into a doughnut. Evaluation will be based on exams and homework. Prerequisite: Mathematics 301, or permission of the instructor and Mathematics 305 or 312. Not open to students who have taken Mathematics 323.