Homotopy
Central Question
When should two continuous maps be considered essentially the same?
Why This Matters
Allowing continuous deformation introduces a coarser notion of equivalence, making loops and holes accessible to algebraic study.
Learning Prompt
Introduce homotopy by asking when continuous maps should be considered equivalent. Develop paths, loops, path homotopy and contractibility. Explain why loops around a hole differ fundamentally from loops inside a disk.