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:

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

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