The derived mappings and the order of a set-valued mapping between topological spaces (Q1380727)

For a multimap \(\phi\) of topological spaces \(X\) and \(Y\), the derived multimap \(\phi'\) of \(\phi\) is defined by the relation \(\phi'(x) = \{y \in \phi(x) \): for each neighbourhood \(U\) of \(y\) there exists a neighbourhood \(V\) of \(x\) such that \(\phi(x') \cap U \neq \emptyset\) for all \(x' \in V\}\). For each ordinal number \(\alpha\), a multimap \(\phi^{(\alpha)}\) is defined by the relations: \(\phi^{(0)} = \phi\), \(\phi^{(\alpha + 1)} = \phi^{(\alpha)}\), \(\phi^{(\alpha)} = \bigcap_{\beta < \alpha}\phi^{(\beta)}\) when \(\alpha\) is a limit ordinal. The order \(\phi\) is defined as \(\min\{\alpha: \phi^{(\alpha + 1)} = \phi^{(\alpha)}\}\). The author studies the relations between topological spaces and the possible orders of multimaps between them. In particular, examples are given demonstrating that every ordinal number is the order of some multimap.