Hyperspaces of continua with connected boundaries in π-Euclidean Peano continua

Let X be a nondegenerate Peano unicoherent continuum. The family CB(X) of proper subcontinua of X with connected boundaries is a G(delta)-subset of the hyperspace C(X) of all subcontinua of X. If every nonempty open subset of X contains an open subset homeomorphic to R-n (such space is called pi-n-Euclidean) and 2 <= n < infinity, then C(X) \ CB(X) is recognized as an F-sigma-absorber in C(X); if additionally, no one-dimensional subset separates X, then the family of all members of CB(X) which separate X is a D-2(F-sigma)-absorber in C(X), where D-2(F-sigma) denotes the small Borel class of differences of two sigma-compacta. (C) 2019 Elsevier B.V. All rights reserved.