Multicoherence of Whitney levels

Abstract

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. (A) If s ⩽ t, then r( μ −1( s))⩾ r( μ −1( t)), 2. (B) r( μ −1( t)) is finite for every t > 0, 3. (C) if 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 4. (D) X is a simple closed curve if and only if r( μ −1( t)) > 0 for every t < 1.