More absorbers in hyperspaces

TL;DR

This paper investigates the complexity and absorption properties of various families of subcontinua within hyperspaces of manifolds and Hilbert cube manifolds, revealing their classification in descriptive set theory.

Contribution

It characterizes multiple families of subcontinua as absorbers in hyperspaces, establishing their descriptive set-theoretic complexity and topological universality.

Findings

Families of separating subcontinua are $F_\sigma$-absorbers.

Certain continua families are $D_2(F_\sigma)$-absorbers.

Hereditarily infinite-dimensional compacta are $\Pi^1_1$-complete.

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 $2^{I^\omega}$.