Topology

• Topology and continuity: bases, order topology, subspace topology, product topology (infinite and finite), box topology, closed sets, limit points, Hausdorff spaces, homeomorphisms, Pasting Lemma, metric spaces, uniform topology, Uniform Limit Theorem, quotient topology, open and closed maps.
• Compactness and connectedness: connectedness of R, Intermediate Value Theorem, connectedness for products, path-connectedness, components and path-components, Tube Lemma, finite intersection property, uniform continuity, Heine-Borel Theorem, Lebesgue Number Lemma, sequential compactness, limit point compactness, local compactness, compactifications.
• Complete metric spaces: Cauchy sequences, equicontinuity, Ascoli Theorem (for Rⁿ), complete and totally bounded metric spaces.
• Definition and elementary properties of homotopy; homotopy equivalences; deformation retracts.
• The definition of the fundamental group π₁; functoriality under mappings and invariance under homotopy. The relation between π₁ at different base points. The fundamental group of a cartesian product.
• The path lifting/homotopy lifting lemmas, their proofs, and their use in proving that π₁(S¹)≅Z.
• The statement of the Seifert–van Kampen theorem, and its use in computing π₁ of various spaces, such as compact surfaces.
• Covering spaces; path and homotopy lifting theorems; classification of connected covers via subgroups of the fundamental group.
• Cell complexes, ∆-complexes and simplicial complexes, the classification of compact surfaces.
• Singular, simplicial and cellular homology; Hurewicz theorem relating H₁ and π₁; degree of maps between spheres (and connected orientable manifolds), induced homomorphisms, homotopy invariance; reduced homology; relative homology; long exact sequences of a pair, a triple, and the Mayer-Vietoris sequence; excision; Homology with coefficients, the universal coefficients theorem; Euler characteristic.
• Simplicial, singular and cellular cohomology; the cup product; Künneth theorems; orientations, the cap product and Poincaré duality

References

• Munkres, Topology: A First Course, Sections 1-7, 12-29, 43, 45
• Hatcher, Algebraic Topology, Chapters 0, 1, 2, 3.