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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on continuous mappings and the property of J. L. Kelley


Author: Hisao Kato
Journal: Proc. Amer. Math. Soc. 112 (1991), 1143-1148
MSC: Primary 54B20; Secondary 54C05, 54C60, 54C65
MathSciNet review: 1073527
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Abstract: In this paper, it is proved that if $ X$ is a continuum and $ \omega $ is any Whitney map for $ C(X)$, then the following are equivalent:

(1) $ X$ has property [K].

(2) There exists a (continuous) mapping $ F:X \times I \times [0,\omega (X)] \to C(X)$ such that $ F(\{ x\} \times I \times \{ t\} ) = \{ A \in {\omega ^{ - 1}}(t)\vert x \in A\} $ for each $ x \in X$ and $ t \in [0,\omega (X)]$, where $ I = [0,1]$.

(3) For each $ t \in [0,\omega (X)]$, there is an onto map $ f:X \times I \to {\omega ^{ - 1}}(t)$ such that $ f(\{ x\} \times I) = \{ A \in {\omega ^{ - 1}}(t)\vert x \in A\} $ for each $ x \in X$. Some corollaries are obtained also.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1073527-4
PII: S 0002-9939(1991)1073527-4
Keywords: Hyperspaces of continua, Whitney map, property $ [{\text{K}}]$, continuous selection, weakly chainable, uniformly pathwise connected
Article copyright: © Copyright 1991 American Mathematical Society