This report treats a few selected topics from Riemannian geometry and algebraic topology. It consists of three parts. The rst part discusses basic homotopy theory, until the covering spaces and the Seifert-van Kampen theorems. As another topic, smooth manifolds and their CW-structure are explored. The second part outlines a fairly new discipline, algebraic ditopology, arriving at a discussion of the fundamental category for ordered topological spaces. The third part contains a thorough treatment of Riemannian geometry, nishing with the Hopf-Rinow theorem

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