The hyperspaces of hereditarily decomposable continua and of decomposable subcontinua without pseudoarcs in the cube of dimension greater than 2 are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval [0, 1]. Moreover, in such a cube, all indecomposable subcontinua form a homotopy dense subset of the hyperspace of (nonempty) subcontinua.

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 .

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