Topology is sometimes referred to as 'rubber-sheet geometry'. It studies continuity in its broadest context. We begin by analysing the notion of continuity familiar from calculus, showing that it depends on being able to measure distance between points in Euclidean space. This leads to the more general notion of a metric space. While metric spaces have many important applications, we see that they do not provide the most suitable context for studying continuity. A deeper analysis of continuity in metric spaces leads us to generalise the topological spaces, which provide the broadest setting for continuity.The central concepts of topology, compactness and connectedness, are introduced and applied to prove such central results in mathematics as the Fundamental Theorem of Algebra, the Extreme Value Theorem, the Intermediate Value Theorem. These show the relationship between modern advanced mathematics and the senior school mathematics curriculum. Applications of topology to number theory, algebraic geometry and functional analysis are featured.Since metric spaces are important in geometric contexts, these concepts are applied to them, and the notion of completeness is introduced. The Banach Fixed Point Theorem, important for differential equations and Newton's Method is also proved.The presentation follows modern developments and, in addition this unit provides the basis for studying differential geometry, functional analysis, classical and quantum mechanics, dynamical systems, algebraic and differential topology. This unit is essential for students wishing to continue to honours or post-graduate mathematics.