1. Perverse sheaves: https://en.wikipedia.org/wiki/Perverse_sheaf
2. Blowing up points on a plane: https://en.wikipedia.org/wiki/Blowing_up
3. Balls?... https://en.wikipedia.org/wiki/Ball_(mathematics)#In_topological_spaces
4. Hairy ball theorem: https://en.wikipedia.org/wiki/Hairy_ball_theorem
5. Tropical geometry: https://en.wikipedia.org/wiki/Tropical_geometry
6. Radicals: https://en.wikipedia.org/wiki/Radical_of_an_ideal
7. Cardinals: https://en.wikipedia.org/wiki/Cardinal_number
8. Sheets: https://en.wikipedia.org/wiki/Covering_space#Formal_definition or https://en.wikipedia.org/wiki/Riemann_surface
9. Twin primes: https://en.wikipedia.org/wiki/Twin_prime
10. Necklace splitting theorem: https://en.wikipedia.org/wiki/Necklace_splitting_problem