Spaces of Whitney maps

Alejandro Illanes

Vol. 139 (1989), No. 1, 67–77

DOI: 10.2140/pjm.1989.139.67

Abstract

Let X be a continuum. Let 2^X (respectively, C(X)) be the hyperspace of nonempty closed subsets (respectively, subcontinua) of X, endowed with the Hausdorff metric. For 𝒦 = C(X) or 2^X, let W(𝒦) denote the space of Whitney maps for 𝒦 with the “sup metric” and pointwise product. In this paper we prove that if there exists a homeomorphism ϕ : W(C(X)) → W(C(Y )) (or ϕ : W(2^X) → W(2^Y )) which preserved products and “strict order”, then X is homeomorphic to Y . We also prove that there exists an embedding ψ : W(C(X)) → W(2^X) such that ψ(u) is an extension of u for each u ∈ W(C(X)).

Mathematical Subject Classification 2000

Primary: 54B20

Secondary: 54C35, 54F15

Milestones

Received: 25 October 1987

Revised: 24 May 1988

Published: 1 September 1989