Topology

When introducing topological spaces, most books start with the axioms, which look abstract at first sight. Here, the axioms are motivated through the notion of continuity on the real line, which is a familiar notion. The concept of continuity is then generalised to metric spaces and finally to topological spaces, which is the home ground of the concept (one view is that topology is a study of continuous functions). Beginning with preliminaries and basic concepts of open sets, limit points, and closed sets, the discussion progresses to Hausdorff spaces and second countable spaces. Finally, homeomorphism and the topological properties of connectedness, homotopy—simply and multiply connected spaces—compactness are described in detail.