- Simplicial complexes from a combinatorial and geometric view; subcomplexes, filtrations (incl. Vietoris–Rips & Čech), geometric realisation, simplicial maps, barycentric subdivision, and local geometry via stars/links/cones. - Homotopy: Homotopies of maps and spaces; contractibility; carriers & Carrier Lemma; Quillen’s fiber theorem; nerve theorem (cover → nerve equivalence); elementary collapses; simplicial approximation. - Homology: Euler characteristic; orientations/boundaries; chain complexes; (co)homology groups; Smith decomposition. - Sequences: Categorical/chain-map viewpoint; functoriality; chain homotopy; snake lemma; relative homology; Mayer–Vietoris; homotopy invariance. - Cohomology: Cochains & cohomology; cup/cap products; Poincaré duality; (bonus) Künneth formula. - Persistence: Persistent homology, barcodes, matrix-reduction algorithm for filtrations, interleaving distance, stability theorem. - Sheaves: Sheaves and (co)homology from a TDA angle; étale spaces; pushforward/pullback; (bonus) cosheaves. - Gradients: Discrete Morse theory for complexes, persistence, and sheaves: acyclic matchings, Morse chain complex, equivalence results.