Topological spaces, metric spaces and continuous maps: definitions, general properties, examples. Subspaces. Finite products of topological spaces. Quotient topological spaces: examples (projective spaces, spheres, tori,...). Hausdorff spaces. Connected spaces and path-connected spaces, compact spaces, examples: relation with some concepts of Analysis. The theorem of Tychonoff. Locally compact spaces and the compactification theorem of Alexandroff. Locally connected spaces. Complete metric spaces. Urysohn's lemma and Urysohn's metrizability theorem. Tietze's extension theorem. Baire spaces.