## Retracts in metric spaces

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- by Lech Pasicki
- Proc. Amer. Math. Soc.
**78**(1980), 595-600 - DOI: https://doi.org/10.1090/S0002-9939-1980-0556639-3
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## Abstract:

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 $S:X \times \langle 0,1\rangle \times X (x,\alpha ,y) \mapsto {S_x}(\alpha ,y) \in X$ 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.

## References

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## Bibliographic Information

- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**78**(1980), 595-600 - MSC: Primary 54C15
- DOI: https://doi.org/10.1090/S0002-9939-1980-0556639-3
- MathSciNet review: 556639