Hyperspaces of continua with connected boundaries in $\pi$-Euclidean Peano continua

TL;DR

This paper studies the topological structure of hyperspaces of continua with connected boundaries within $ ext{pi}$-Euclidean Peano continua, revealing their absorber properties in the hyperspace of all subcontinua.

Contribution

It characterizes the hyperspaces of continua with connected boundaries as $G_\delta$-sets and identifies their absorber properties in $ ext{pi}$-Euclidean Peano continua.

Findings

$CB(X)$ is a $G_\delta$-subset of $C(X)$.

$C(X) ackslash CB(X)$ is an $F_\sigma$-absorber.

The family of boundary-connected continua that separate $X$ is a $D_2(F_\sigma)$-absorber.

Abstract

Let $X$ be a nondegenerate Peano unicoherent continuum. The family $CB(X)$ of proper subcontinua of $X$ with connected boundaries is a $G_\delta$-subset of the hyperspace $C(X)$ of all subcontinua of $X$. If every nonempty open subset of $X$ contains an open subset homeomorphic to $\mathbb R^n$ (such space is called $\pi$-$n$-Euclidean) and $2\le n<\infty$, then $C(X)\setminus CB(X)$ is recognized as an $F_\sigma$-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 $D_2(F_\sigma)$-absorber in $C(X)$, where $D_2(F_\sigma)$ denotes the small Borel class of differences of two $\sigma$-compacta.