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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Shape properties of Whitney maps for hyperspaces


Author: Hisao Kato
Journal: Trans. Amer. Math. Soc. 297 (1986), 529-546
MSC: Primary 54B20
DOI: https://doi.org/10.1090/S0002-9947-1986-0854083-2
MathSciNet review: 854083
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Abstract: In this paper, some shape properties of Whitney maps for hyperspaces are investigated. In particular, the following are proved:

(1) Let $ X$ be a continuum and let $ \mathfrak{H}$ be the hyperspace $ {2^X}$ or $ C(X)$ of $ X$ with the Hausdorff metric. Then if $ \omega $ is any Whitney map for $ \mathfrak{H}$, for any $ 0 \leqslant s \leqslant t \leqslant \omega (X){\omega ^{ - 1}}(t)$ is an approximate strong deformation retract of $ {\omega ^{ - 1}}([s,t])$. In particular, $ \operatorname{Sh} ({\omega ^{ - 1}}(t)) = \operatorname{Sh} ({\omega ^{ - 1}}([s,t]))$.

(2) Pointed $ 1$-movability is a Whitney property.

(3) For any given $ {\text{n}} < \infty $, the property of (cohomological) dimension $ \leqslant n$ is a sequential strong Whitney-reversible property.

(4) The property of being chainable or circle-like is a sequential strong Whitney-reversible property.

(5) The property of being an FAR is a Whitney property for $ 1$-dimensional continua.

Property (2) is an affirmative answer to a problem of J. T. Rogers [16, 112]. Properties (3) and (4) are affirmative answers to problems of S. B. Nadler [20, (14.57) and 21].


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DOI: https://doi.org/10.1090/S0002-9947-1986-0854083-2
Keywords: Hyperspaces, Whitney maps, Whitney property, (sequential) strong Whitney-reversible property, approximate strong deformation retract, pointed $ 1$-movable, FAR, ANR, chainable, circle-like, $ \varepsilon $-mapping, indecomposable
Article copyright: © Copyright 1986 American Mathematical Society