0 Introduction. - Notation. - Brouwer Fixed Point Theorem. - Categories and Functors. - 1. Some Basic Topological Notions. - Homotopy. - Convexity Contractibility and Cones. - Paths and Path Connectedness. - 2 Simplexes. - Affine Spaces. - Affine Maps. - 3 The Fundamental Group. - The Fundamental Groupoid. - The Functor ?1. - ?1(S1). - 4 Singular Homology. - Holes and Green's Theorem. - Free Abelian Groups. - The Singular Complex and Homology Functors. - Dimension Axiom and Compact Supports. - The Homotopy Axiom. - The Hurewicz Theorem. - 5 Long Exact Sequences. - The Category Comp. - Exact Homology Sequences. - Reduced Homology. - 6 Excision and Applications. - Excision and Mayer-Vietoris. - Homology of Spheres and Some Applications. - Barycentric Subdivision and the Proof of Excision. - More Applications to Euclidean Space. - 7 Simplicial Complexes. - Definitions. - Simplicial Approximation. - Abstract Simplicial Complexes. - Simplicial Homology. - Comparison with Singular Homology. - Calculations. - Fundamental Groups of Polyhedra. - The Seifert-van Kampen Theorem. - 8 CW Complexes. - Hausdorff Quotient Spaces. - Attaching Cells. - Homology and Attaching Cells. - CW Complexes. - Cellular Homology. - 9 Natural Transformations. - Definitions and Examples. - Eilenberg-Steenrod Axioms. - Chain Equivalences. - Acyclic Models. - Lefschetz Fixed Point Theorem. - Tensor Products. - Universal Coefficients. - Eilenberg-Zilber Theorem and the Künneth Formula. - 10 Covering Spaces. - Basic Properties. - Covering Transformations. - Existence. - Orbit Spaces. - 11 Homotopy Groups. - Function Spaces. - Group Objects and Cogroup Objects. - Loop Space and Suspension. - Homotopy Groups. - Exact Sequences. - Fibrations. - A Glimpse Ahead. - 12 Cohomology. - Differential Forms. - Cohomology Groups. - Universal Coefficients Theorems for Cohomology. -Cohomology Rings. - Computations and Applications. - Notation. Language: English