In this paper we define S-contractibility and two classes of spaces connected with this notion. A space X is said to be S-contractible provided that S is a function S : X × ⟨ 0 , 1 ⟩ × X ( x , α , y ) ↦ S x ( α , y ) ∈ X S:X \times \langle 0,1\rangle \times X (x,\alpha ,y) \mapsto {S_x}(\alpha ,y) \in X that is continuous in α \alpha and y, and for every x , y ∈ X , S x ( 0 , y ) = y , S x ( 1 , y ) = x x,y \in X,{S_x}(0,y) = y,{S_x}(1,y) = x . This notion is close to equiconnectedness, which can be defined as follows. A space X is equiconnected if there exists a map S such that X is S-contractible and S x ( α , x ) = x {S_x}(\alpha ,x) = x for all x ∈ X x \in X and α ∈ I \alpha \in I (cf. [4]). The results we obtain in the theory of retracts are close to those that are known for equiconnected spaces. Also the thickness of the neighborhood that can be retracted on a set in a metric space is estimated, which enables to prove a theorem belonging to fixed point theory.