Cells in $n$-fold hyperspaces
Volume 152 / 2018
Colloquium Mathematicum 152 (2018), 45-53
MSC: Primary 54B20; Secondary 54F15.
DOI: 10.4064/cm7223-9-2017
Published online: 22 January 2018
Abstract
Given a metric continuum $X$, let $C_{n}(X)$ denote the hyperspace of nonempty closed subsets of $X$ with at most $n$ components. A $k$-od in $X$ is a subcontinuum $B$ of $X$ which contains a subcontinuum $A$ such that $B\setminus A$ has at least $k$ components. We prove that if $1\leq n\leq m$, then $C_{n}(X)$ contains an $m$-cell if and only if there exist positive integers $k_{1},\ldots ,k_{n}$ and pairwise disjoint subcontinua $B_{1},\ldots ,B_{n}$ of $X$ such that for each $i$, $B_{i}$ is a $k_{i}$-od in $X$ and $k_{1}+\cdots +k_{n}=m$.
