Multicoherence of Whitney levels

Let X be a Peano continuum and μ : C(X) → [0, 1] a Whitney map with C(X) the hyperspace of subcontinua of X. For a connected space Y, let r(Y) denote the multicoherence degree of Y. In this paper we prove: 1. s ⩽ t, then r(μ−1(s))⩾ r(μ−1(t)), μ−1(t)) is finite for every t > 0, 0 < m ⩽ r(X), then there exists a Whitney map ν : C(X) → [0,1] and there exists t ϵ [0,1] such that r(ν−1(t)) = m, and is a simple closed curve if and only if r(μ−1(t)) > 0 for every t < 1.