Retracts in metric spaces

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 $S:X \times \langle 0,1\rangle \times X \mathrel (x,\alpha ,y) \mapsto {S_x}(\alpha ,y) \in X$ that is continuous in $\alpha$ and y, and for every $x,y \in X,{S_x}(0,y) = y,{S_x}(1,y) = x$. 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 ${S_x}(\alpha ,x) = x$ for all $x \in X$ and $\alpha \in I$ (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.