Homotopy

A homotopy between continuous maps $f,g:X\to Y$ is a continuous map $H:X\times [0,1]\to Y$ satisfying $H(x,0)=f(x)$ and $H(x,1)=g(x)$. Think of $t$ as time: continuity is required jointly in position and time. The target $Y$ specifies where every intermediate point is allowed to go. See Hatcher, Algebraic Topology, Chapter 0, pp. 3–4.

For the filled disk $D^2=\{x\in {\mathbb{R}}^2:\Vert x\Vert \le 1\}$, 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 $\Vert H(x,t)\Vert =(1-t)\Vert x\Vert \le 1$, so the formula never leaves the disk. For $x=(3/5,4/5)$, the midpoint of the deformation is $(3/10,2/5)$, whose Euclidean norm is $1/2$.

On the circle $S^1=\{x:\Vert x\Vert =1\}$, the same formula fails: at $t=1/2$ every image has Euclidean norm $1/2$, so lies outside $S^1$. 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.

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.