Sudden homotopy

Definition
A sudden homotopy on a topological space $$X$$ is a continuous map $$F:X \times I \to X$$ such that $$F(x,0) = x \ \forall \ x$$ and for any $$t < 1$$ the map $$x \mapsto F(x,t)$$ is a homeomorphism from $$X$$ to $$X$$.

Of particular interest is sudden contracting homotopy which is both sudden and contracting.