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

Abstract

Let X be a nondegenerate Peano unicoherent continuum. The family CB(X) of proper subcontinua of X with connected boundaries is a G δ -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 π-n-Euclidean) and 2 ≤ n < ∞, then C(X) \ CB(X) is recognized as an F σ -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 σ )-absorber in C(X), where D 2 (F σ ) denotes the small Borel class of differences of two σ-compacta.All continua in the paper are metric. For a continuum X, we consider the hyperspaces 2 X = {A ⊂ X : A is closed and nonempty} and C(X) = {A ∈ 2 X : A is connected} with the Hausdorff metric. Definewhere Bd(A) denotes the boundary of A in X. Evaluation of the Borel complexity of CB(X)K. Kuratowski observed in [7, p. 156] that, for any compact nondegenerate X, the function α : 2 X \ {X} → 2 X , α(A) = X \ A is lower semicontinuous, hence of the first Borel class, while the function Bd : 2 X \ {X} → 2 X , A → Bd(A) = A ∩ X \ A, is of the second Borel class for any nondegenerate continuum X. It means that, for each nondegenerate continuum X, the preimage α −1 (D) of a closed

The strong universality of ANRs with a suitable algebraic structure

“…A topological semilattice X is called a Lawson semilattice if it has a base of the topology consisting of subsemilattices. Lawson semilattices often appear in the theory of hyperspaces, see [20], [14], [15], [16], [19], [17], [18].…”

“…Theorem 10 has been applied in [6] to studying the topological structure of some hyperspaces. For more applications of strongly universal and absorbing spaces in the theory of hyperspaces, see [7], [10], [14], [15], [16], [19], [17], [18].…”