Induced mappings on the hyperspace of totally disconnected sets (Q6610436)

For a Hausdorff space \(X\), let \(K(X)\) denote the hyperspace of all nonempty compact subsets of \(X\) endowed with the Vietoris topology and \(TD(X)\) the subspace of \(K(X)\) consisting of all totally disconnected subspaces of \(X\). In the sequel, any mapping is assumed to be continuous. A mapping \(f \colon X \to Y\) between Hausdorff spaces induces a mapping \(TD(f) \colon TD(X) \to K(X)\) by letting \(TD(f) (A) = f(A) \, (=\{ f(a) : a \in A\})\) for \(A \in TD(X)\). Let \(\mathbb{M}\) be a class of mappings between Hausdorff spaces. In this paper, the authors study the interrelation between the condition ``\(f \in \mathbb{M}\)'' and the condition ``\(TD(f) \in \mathbb{M}\)'' for many classes \(\mathbb{M}\).\N\NLet \(f \colon X \to Y\) be a mapping between Hausdorff spaces. The authors first prove that \(TD(f)(A) \in TD(Y)\) for each \(A\in TD(X)\) if and only if \(f\) is co-monotone, i.e., for any \(A \in K(X)\), if \(f(A)\) is a non-degenerate connected subset of \(Y\), then \(A\) has a non-degenerate component. Then the following is proved: Suppose that \(\mathbb{M}\) is a class of all mappings satisfying one of the following conditions: open; almost-open; pseudo-open; closed; quotient; semi-open; dense-open; almost-interior; monotone; confluent; light. If \(f\) is a co-monotone mapping and \(TD(f) \in \mathbb{M}\), then \(f \in \mathbb{M}\).\N\NMoreover, the authors give examples of co-monotone mappings satisfying \(f \in \mathbb{M}\) and \(TD(f) \notin \mathbb{M}\) for a class \(\mathbb{M}\) of all mappings satisfying one of the following conditions: open; almost-open; pseudo-open; quotient; closed.