An Introduction to Metric Spaces

Sammanfattning: In this thesis we start off by ensuring that the reader is up to speed when it comes to some well known definitions and theorems from real analysis. We then introduce the reader to metric spaces and provide them with some examples such as the real numbers with the Euclidean distance, and compact sets with the Hausdorff distance. Then, we go on to define important concepts such as inner points, limit points, open sets, boundary and much more. We also show, whenever we can, how these concepts are connected. With these tools in place we move on to explain how limits and continuity are defined in metric spaces as well as providing the reader with several examples. We then introduce the reader to the concepts of compactness and uniform convergence, for which we show some interesting results such as how uniform convergence and the supremum norm are related. We finish off by covering curves and connectedness (including pathconnectedness) in metric spaces, before we briefly touch on topological spaces as to give the reader a hint of what further mathematics studies might hold.