Weakly contractible not implies contractible

This article gives the statement and possibly, proof, of a non-implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property need not satisfy the second topological space property
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Statement

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

Definitions used

Weakly contractible space

Further information: 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

Further information: contractible space

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

Proof

Further information: double comb space

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. Fill this in later