= Weakly contractible space
{title2=$\pi_n(X)=0\ (n\ge1)$}
A <weakly contractible space> is a nonempty <path-connected> <topological space> $X$ with trivial <homotopy groups> $\pi_n(X,x)$ for every $n\ge1$. Equivalently, the map $X\to\{*\}$ is a <weak homotopy equivalence>. For a <CW complex>, the <Whitehead theorem> then implies that $X$ is <contractible>.<Algebraic Topology by Allen Hatcher>[section 4.1, Whitehead's theorem]{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.
Back to article page
