Weakly contractible space (${\pi }_n(X)=0(n\ge 1)$)

A weakly contractible space is a nonempty path-connected topological space $X$ with trivial homotopy groups ${\pi }_n(X,x)$ for every $n\ge 1$. Equivalently, the map $X\to \{\ast \}$ is a weak homotopy equivalence. For a CW complex, the Whitehead theorem then implies that $X$ is contractible.^[ref]

Every contractible space is weakly contractible: a contraction makes every based sphere map nullhomotopic. The real line is a simple example, contracted by $H(x,t)=(1-t)x$. In contrast, the circle fails the test because its fundamental group is $\mathbb{Z}$. For arbitrary topological spaces, weak contractibility alone does not guarantee a continuous contraction; the CW complex hypothesis in the stated implication supplies substantive control over the space.