Arc-smoothness and contractibility in Whitney levels

Abstract

Let X X be a continuum. Let 2 x {2^x} (resp., C ( X ) C(X) ) be the space of all nonempty closed subsets (resp., subcontinua) of X X . In this paper we prove that if X X is an arc-smooth continuum, then there exists an admissible Whitney map μ : 2 x → R \mu :{2^x} \to {\mathbf {R}} such that μ | C ( X ) : C ( X ) → R \mu |C(X):C(X) \to {\mathbf {R}} is admissible and for every t ∈ ( 0 , μ ( X ) ) , μ − 1 ( t ) t \in (0,\mu (X)),{\mu ^{ - 1}}(t) and ( μ | C ( X ) ) − 1 ( t ) {(\mu |C(X))^{ - 1}}(t) are arc-smooth. This answers a question by J. T. Goodykoontz, Jr. Also we give an example of a contractible continuum X X such that, for every Whitney map υ : C ( X ) → R \upsilon :C(X) \to {\mathbf {R}} there exists t ∈ ( 0 , υ ( X ) ) t \in (0,\upsilon (X)) such that υ − 1 ( t ) {\upsilon ^{ - 1}}(t) is not contractible.