The Fundamental Theorem of Homotopy

The Fundamental Theorem of Homotopy states that, for certain spaces, continuous transformations (homotopies) that start and end at the same points can be classified by their overall shape or structure. Essentially, it means that if two paths or shapes can be smoothly deformed into each other without breaking, then they are considered equivalent in a fundamental way. This helps mathematicians understand how spaces can be connected or related through continuous changes, providing a powerful way to classify and study topological spaces based on their inherent properties rather than exact geometric details.