Metric Spaces (University of Groningen)
This is a required course for University of Groningen students in the programs BSc Mathematics, BSc Applied Mathematics, BSc Physics & Mathematics and an elective course in the MSc Physics and Applied Physics. I taught this course from 2008 to 2012.
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.