Contracting homotopy

Definition

Let $X$ be a topological space. A contracting homotopy for $X$ is a homotopy $F:X\times I\to X$ such that there exists a point $x_0\in X$ such that:

• $F(x,0)=x\mathrm{\forall }x\in X$
• $F(x,1)=x_0\mathrm{\forall }x\in X$

A topological space which has a contracting homotopy is termed a contractible space.