More absorbers in hyperspaces

Abstract

The family of all subcontinua that separate a compact connected $n$-manifold $X$ (with or without boundary), $n\ge 3$, is an $F_{\sigma }$-absorber in the hyperspace $C(X)$ of nonempty subcontinua of $X$. If $D_2(F_{\sigma })$ is the small Borel class of spaces which are differences of two $\sigma$-compact sets, then the family of all $(n-1)$-dimensional continua that separate $X$ is a $D_2(F_{\sigma })$-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 $F_{\sigma \delta }$-absorbers in the hyperspace $C(I^n)$ for $n\ge 3$. 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 ${\Pi }_1^1$-absorbers in $C(X)$. The family of all hereditarily infinite-dimensional compacta in the Hilbert cube $I^{\omega }$ is ${\Pi }_1^1$-complete in $2^{I^{\omega }}$.

References in corpus (1)

• Wilder continua and their subfamilies as coanalytic absorbers

Cited by in corpus (5)

• Wilder continua and their subfamilies as coanalytic absorbers
• Hyperspaces of continua with connected boundaries in $\pi$-Euclidean Peano continua
• The hyperspace of non-cut subcontinua of graphs and dendrites
• The strong universality of ANRs with a suitable algebraic structure
• Hyperspaces of countable compacta