This is a compulsory course for Bachelor's students in the programs Mathematics, Technical Mathematics, and Physics & Mathematics and a selective course for MSc students in the programs Physics and Applied Physics.
In Euclidean spaces, continuity is defined using the familiar Euclidean distance between points. In this course we generalize the concept of continuity first to metric and then to topological spaces. In metric spaces we are given the concept of distance between its “points”. Function spaces are some of the most useful examples of metric spaces. In topological spaces we know only which subsets are open. It turns out that this is all we need in order to study continuity. We discuss further some important analytical and geometrical concepts such as compactness, connectedness, and completeness.
- Introduction to metric and topological spaces, W.A. Sutherland, Oxford University Press, 2nd edition, 2009.