Cells in hyperspaces
Let \(C(X)\) denote the hyperspace of subcontinua of a continuum \(X\) equipped with the Hausdorff metric. For \(A,B\in C(X)\) with \(B\subset A\), \(C(B, A)=\{D\in C(A): B\subset D\}\), and if \(p\in X\), \(C(\{p\}, X)\) is denoted by \(C(p, X)\). For a natural number \(k\) , a continuum \(Y\) is called a \(k\)-od, if there exists \(M\in C(Y)\) such that \(Y\setminus M\) has at least \(k\) components, and this \(M\) is called the core of the \(k\)-od. A space \(X\) is called a \(k\)-cell, if \(X\) is homeomorphic to \([0,1]^k\). The author proves the following: Theorem 3.11. For a natural number \(k> 1\) and \(p\in X\), if \(p\) is not in the core of any \(k\)-od, then \(C(p, X)\) contains no \(k\)-cells. Corollary 3.12. The point \(p\in X\) is contained in the core of a \(k\)-cell if and only if \(C(p, X)\) contains a \(k\)-cell. Corollary 3.15. Let \(X\) be a continuum and \(L\in C(X)\). Then \(L\) is contained in the core of a \(k\)-od if and only if \(C(L,X)\) contains a \(k\)-cell. As a consequence, the author shows: For a continuum \(X\) and \(A\in C(X)\), \(A\) is contained in the core of a \(k\)-od if and only if \(C(A, X)\) contains a \(k\)-cell.
scientific article; zbMATH DE number 5131629