Classically, mathematics is divided into three main branches: geometry, algebra and analysis. The purpose of this two-semester course is to provide an introduction to topology, which is at the foundations of the branches geometry and analysis. Roughly speaking, topology is the study of mathematical properties that are invariant under continuous operations.
The basic concepts of point-set topology will first be introduced in the context of metric spaces, for which there is a notion of distance. But that will be only a pedagogical prelude to the notion of topological space, of which metric spaces are but a special case. The familiar notions of convergence and continuity will be reviewed in this general setting, and then the standard topics of connectedness, compactness, product and quotient spaces, separation properties and metrization will be discussed.
Second semester: MTG 4303/5317 Introduction to Topology, II: After this lengthy introduction to point-set topology, attention will be turned to algebraic topology, whose purpose is to describe topological properties using algebraic methods, particularly using groups or rings. It will therefore be necessary, in the second semester, that the student already be familiar with these algebraic notions. Chapter 9 of the text will be supplemented by outside material.
A second purpose of this course will be to illustrate and strengthen the student's grasp of the powerful, abstract mode of thought of modern mathematics, and also to improve her/his proof-writing skills.