Shape properties of Whitney maps for hyperspaces

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].