Product topology in the hyperspace of subcontinua
Let \(X\) be a metrizable continuum with its hyperspace of nondegenerate subcontinua \(C(X)\). If \(A\) and \(B\) are (possibly) empty finite subsets of \(X\) then \(U(A,B)=\{K\in C(X) : A\subseteq K\) and \(K\cap B=\emptyset\}\). The collection of all sets of the form \(U(A,B)\) serves as the basis of a topology on \(C(X)\), called the product topology \(\tau_P\) by the authors. Let \(\tau_H\) denote the usual Vietoris topology on \(C(X)\). The authors prove that the topological properties of a dendroid \(X\) are close to those of its hyperspace \(C(X)\) endowed with the product topology. They prove for a dendroid \(X\) that \(((C(X),\tau_p)\) has countable \(\pi\)-weight if and only if \(X\) has countably many endpoints only. In addition, the product topology on \(C(X)\) is separable if and only if every pairwise disjoint family of arcs in \(X\) is countable, and \(\tau_H\subseteq \tau_P\) if and only if \(X\) is a dendrite.
scientific article; zbMATH DE number 1501778