This video is the last in this series on the History of Mathematics, and gives a brief introduction to Topology. The subject goes back to Euler (as do so many things in modern mathematics) with his discovery of the Euler characteristic of a polyhedron, although arguably Descartes had found something close to this in his analysis of curvature of a polyhedron. We introduce this via rational turn angles, a renormalization of angle where a full turn has the value one (very reasonable, and ought to be used more!!) The topological nature of the Euler characteristic was perhaps first understood by Poincare, and we sketch his argument for its invariance under continuous transformations.We discuss the sphere, torus, genus g surfaces and the classification of orientable, and non-orientable closed 2 dimensional surfaces, such as the Mobius band (which has a boundary) and the projective plane (which does not). The interest in these objects resulted from Riemann's work on surfaces associated to multi-valued functions in the setting of complex analysis. Finally we briefly mention the important notion of a simply connected space, and the Poincare conjecture, solved recently by G. Perelman.If you enjoy this subject, you can have a look at my video series Algebraic Topology. Thanks for watching the series!