Concerning hyperspaces of certain Peano continua and strong regularity of Whitney maps

Hisao Kato

Vol. 119 (1985), No. 1, 159–167

DOI: 10.2140/pjm.1985.119.159

Abstract

Let X be a Peano continuum and let H = 2^X (resp., C(X)) be the space of all nonempty closed subsets (resp., subcontinua) of X with Hausdorff metric. If H = C(x), assume that X contains no free arc. Then the following are shown.

1. If ω is an admissible Whitney map for H, then

   is a trivial bundle map with Hilbert cube fibers.

2. If X is the Hilbert cube Q, then there is a strongly admissible Whitney map ω for H such that ω|ω⁻¹([0,ω(X))) → [0,ω(X)) is a trivial bundle map with Hilbert cube fibers.
3. If X is the n-sphere Sⁿ (n = 1,2,…,), then there is a Whitney map ω for 2^X such that for some t₀ ∈ (0,ω(X)), ω|ω⁻¹((0,t₀]) : ω⁻¹((0,t₀]) → (0,t₀] is a trivial bundle map with X × Q fibers. If X is the n-sphere Sⁿ (n = 2,3,…,), there is a Whitney map ω for C(X) such that for some t₀ ∈ (0,ω(X)), ω|ω⁻¹((0,t₀]) is a trivial bundle map with X × Q fibers.

Mathematical Subject Classification 2000

Primary: 54B20

Secondary: 55R10, 54F25

Milestones

Received: 29 April 1983

Revised: 17 January 1984

Published: 1 September 1985