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