Today, topological structures play a crucial role in all areas of mathematics. Starting from the concept of distance, metric spaces are first introduced, which specifically include normed vector spaces. They motivate the concept of the topological space with the associated homomorphisms, namely the continuous mappings. Relevant constructions such as image and pre-image topologies with their universal properties, especially quotients and products, are discussed in detail. The fundamental concepts of connection and compactness, which already played an important role for the spaces \(\mathbb{R}\) and \(\mathbb{C}\) , are central objects of consideration. The Tychonoff theorem and the completeness of metric spaces as well as the various concepts of convergence in mapping spaces up to the Arzelà-Ascoli theorem are discussed. Finally, summability in Hausdorff abelian topological groups is introduced as a generalization of summability in \(\mathbb{R}\) and \(\mathbb{C}\) .