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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Retracts in metric spaces
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by Lech Pasicki PDF
Proc. Amer. Math. Soc. 78 (1980), 595-600 Request permission

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.
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Additional 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