  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.
