This paper proves the following results: let X be a continuum, let k, m ∈ N, and let B ∈ Cm(X), consider the continuous surjection f
k: C
k(X) → C
k(X). We define the mapping B : C
k(X) → C
k+m(X): by B (A) = f
k(A) B. Then following assertions are equivalent: (1) The hyperspace C
k(X) is g-contractible; (2) For each m ∈ N and for each B ∈ C
m(X) the mapping B is a W-deformation in C
k+m(X); (3) For each m ∈ N there exists B ∈C
m(X) such that the mapping B is a W-deformation in C
k+m(X); (4) There exists m ∈ N such that for each B ∈ C
m(X) the mapping B is a W-deformation in C
k+m(X); (5) There exist m ∈ N and B ∈ C
m(X) such that the mapping B is a W-deformation in C
k+m(X).
