Whitney maps for spaces of embedding hypersurfaces (Q1316905)

Let \(X\) be a metrizable compactum. The space \(\text{exp} X = \{A \subset X\) is nonempty and compact\} with Hausdorff metric is said to be the hyperspace \(\text{exp} X\) of a space \(X\) with metric \(\rho\). A closed subspace \({\mathcal A}\) of \(\text{exp} X\) is called an embedding hyperspace if for every \(A \in \text{exp} X\) such that \(A \supset B\) for some \(B \in {\mathcal A}\) we have \(A \in {\mathcal A}\). \(GX\), the space of all embedding hyperspaces is considered as a subspace of \(\text{exp}^ 2 X\) with the induced metric. A continuous function \(\omega:GX \to R\) that satisfies the following conditions is called a Whitney map for the space of embedding hyperspaces: 1) \(\omega [X] = \{0\}\); 2) if \({\mathcal A}, {\mathcal B} \in GX\) and \({\mathcal A} \subset {\mathcal B} \neq {\mathcal A}\), then \(\omega ({\mathcal A}) < \omega ({\mathcal B})\). It is proved that Whitney maps exist. Theorem 1. Let \(X\) be a metrizable continuum. If \(\omega:GX \to [1;1]\) is a Whitney map, \(\omega/ \omega^{-1}(1,1)\) is a trivial fibering with its own Hilbert cube.