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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Retracts in metric spaces


Author: Lech Pasicki
Journal: Proc. Amer. Math. Soc. 78 (1980), 595-600
MSC: Primary 54C15
MathSciNet review: 556639
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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 \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.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0556639-3
Keywords: Retraction, metric space, contractible set
Article copyright: © Copyright 1980 American Mathematical Society