More absorbers in hyperspaces

Abstract

Abstract. The family of all subcontinua that separate a compact connected n-manifold X (with or without boundary), n ≥ 3, is an Fσ-absorber in the hyperspace C(X) of nonempty subcontinua of X. If D 2 (Fσ) is the small Borel class of spaces which are differences of two σ-compact sets, then the family of all (n−1)-dimensional continua that separate X is a D 2 (Fσ)-absorber in C(X). The families of nondegenerate colocally connected or aposyndetic continua in I n and of at least two-dimensional or decomposable Kelley continua are F σδ -absorbers in the hyperspace C(I n ) for n ≥ 3. The hyperspaces of all weakly infinite-dimensional continua and of C-continua of dimensions at least 2 in a compact connected Hilbert cube manifold X are Π 1 1 -absorbers in C(X). The family of all hereditarily infinite-dimensional compacta in the Hilbert cube I ω is Π 1 1 -complete in 2 I ω .

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].…”

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

“…Hence, there exists a homeomorphism h : I ∞ → C(X) such that h ∂(I ∞ ) = C(X) \ CB(X). It is proved in [6,Theorem 5.8] that S(X)∩N(X)∩C(X) is a D 2 (F σ )absorber in C(X) if X satisfies hypotheses of Theorem 2.3, n ≥ 3 and no subset of dimension ≤ 1 separates X. In a similar way, we will show that S(X) ∩ CB(X) is also a D 2 (F σ )-absorber in C(X) for such X.…”

“…The proof is based on a technique developed in [2,3]. For our purpose, we closely follow its rough description given in [6,Section 3.2]. Without loss of generality, we can assume that ϕ from Proposition 2.4 satisfies Notice that the equivalences remain valid if, given q / ∈ K, we add to the one-dimensional part A(q), appearing in the construction of embedding g(q), finitely many pairwise disjoint sets ϕ U i (q), i < m for some m ∈ N, such that A(q) ∩ ϕ U i (q) is a singleton for each i.…”

“…Without loss of generality, we can assume that ϕ from Proposition 2.4 satisfies Notice that the equivalences remain valid if, given q / ∈ K, we add to the one-dimensional part A(q), appearing in the construction of embedding g(q), finitely many pairwise disjoint sets ϕ U i (q), i < m for some m ∈ N, such that A(q) ∩ ϕ U i (q) is a singleton for each i. It follows that g −1 C(X) \ CB(X) \ K = M \ K. Now, the construction of embedding g, as presented in [6,Section 3.2], satisfies all the required conditions.…”