Homotopy Theory

Homotopy theory is a branch of topology that studies the properties of spaces that are preserved under continuous deformations, such as stretching or bending, but not tearing or gluing. It focuses on the concept of homotopy, which is a way to transform one shape into another gradually. For example, a coffee cup can be continuously transformed into a doughnut shape without cutting. Homotopy theory helps classify spaces based on their "shape" and understand the relationships between them, leading to insights in various fields, including geometry and algebra.