Weakly contractible not implies contractible

Statement
It is possible for a topological space to be a weakly contractible space but not a contractible space.

Weakly contractible space
A topological space $$X$$ is termed weakly contractible if it is a path-connected space and all its homotopy groups are trivial.

Contractible space
A topological space $$X$$ is termed contractible if it is homotopy-equivalent to a point.

Proof
The double comb space is an exampe of a weakly contractible space that is not contractible. This is a subset of the Euclidean plane equipped with the subspace topology.