Connectedness Properties of Whitney Levels

It is shown that δ-connectedness is not a Whitney reversible property. This answers in the negative a question posed by Sam B. Nadler, Jr. in 1978. A topological property P is said to be: (a) a Whitney property provided that if a continuum X has property P, so does μ−1(t) for each Whitney map μ for C(X) and each t ∈ [0, μ(X)) ([6, p. 165]); (b) a Whitney reversible property provided that whenever X is a continuum such that μ−1(t) has property P for all Whitney maps μ for C(X) and all t ∈ (0, μ(X)), then X has property P ([8, p. 235]). A continuum X is said to be: (c) δ-connected provided that for every two points of X there exists an irreducible continuum between them which is hereditarily decomposable ([5, p. 90]); (d) λ-connected provided that for every two points of X there exists an irreducible continuum between them which is of type λ (that is, each of its indecomposable subcontinua has empty interior) ([5, p. 85]). 2000 Mathematics Subject Classification. 54B20, 54F15, 54F50.