This is an introduction to topology for qualified undergraduates.
Metric spaces, topological spaces
Continuous maps and homeomorphisms
Connectedness, compactness (including Heine-Borel, Bolzano-Weierstrass, Ascoli-Arzela theorems),
Fundamental group (homotopy, covering spaces, the fundamental theorem of algebra, Brouwer fixed point theorem)
Surfaces (e.g., Euler characteristic, the index of a vector field, hairy sphere theorem)
Elements of combinatorial topology (graphs and trees, planarity, coloring problems)