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 X be a continuum and let H \mathfrak {H} be the hyperspace 2 X {2^X} or C ( X ) C(X) of X X with the Hausdorff metric. Then if ω \omega is any Whitney map for H \mathfrak {H} , for any 0 ⩽ s ⩽ t ⩽ ω ( X ) ω − 1 ( t ) 0 \leqslant s \leqslant t \leqslant \omega (X){\omega ^{ - 1}}(t) is an approximate strong deformation retract of ω − 1 ( [ s , t ] ) {\omega ^{ - 1}}([s,t]) . In particular, Sh ⁡ ( ω − 1 ( t ) ) = Sh ⁡ ( ω − 1 ( [ s , t ] ) ) \operatorname {Sh} ({\omega ^{ - 1}}(t)) = \operatorname {Sh} ({\omega ^{ - 1}}([s,t])) . (2) Pointed 1 1 -movability is a Whitney property. (3) For any given n > ∞ {\text {n}} > \infty , the property of (cohomological) dimension ⩽ n \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 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].