Homotopy is a concept in topology, a branch of mathematics that studies the properties and structures of spaces that are preserved under continuous transformations. More specifically, homotopy provides a way to classify continuous functions between topological spaces based on their ability to be deformed into one another.
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A homotopy between continuous maps is a continuous map satisfying and . Think of as time: continuity is required jointly in position and time. The target specifies where every intermediate point is allowed to go. See Hatcher, Algebraic Topology, Chapter 0, pp. 3–4.
For the filled disk , the formulais a contraction of a topological space: it starts at the identity map and ends at the constant map to zero. The Euclidean norm obeys , so the formula never leaves the disk. For , the midpoint of the deformation is , whose Euclidean norm is .
On the circle , the same formula fails: at every image has Euclidean norm , so lies outside . This checks one failed candidate; the nonzero winding number of a once-around loop proves that no contraction within the circle exists. The distinction is between allowing the filling and requiring the deformation to stay on its boundary.
