Whitney levels in hyperspaces of non-metrizable continua (Q2254717)

For a metric continuum \(X\) there always exists a Whitney map \(\mu : C(X) \rightarrow [0,1]\) from the hyperspace of subcontinua of \(X\) to the unit interval and sets of the form \(\mu ^{-1}(t)\) for \(t \in [0,1]\) are called Whitney levels. There are non-metrizable continua for which there are no Whitney maps. However the author defines Whitney levels for Hausdorff continua \(X\), even if the hyperspace \(C(X)\) does not necessarily admit Whitney maps. Definition. Let \(X\) be a Hausdorff continuum. Then a Whitney level in \(C(X)\) is a compact subset \(\mathcal W\) of \(C(X)\) such that either \({\mathcal W} = F_{1}(X)\) or \({\mathcal W} \cap F_{1}(X) = \emptyset\) and \(\mathcal W\) satisfies the following two conditions: (1) if \(A, B \in {\mathcal W}\) and \(A \neq B\), then \(A \not\subset B\) and \(B \not\subset A\), (2) \(\mathcal W\) intersects every large order arc in \(C(X)\). (A large order arc is an order arc from a singleton to \(X\).) It is proved that if there is a partition of \(C(X)\) in Whitney levels, then there exists a Whitney map such that its fibers are the elemets of the partition. Next the author studies the particular case when \(X\) is a generalized arc. For each Hausdorff continuum \(X\) two levels in \(C(X)\), called trivial Whitney levels, \(F_{1}(X)\) and \(\{X\}\) always exists. It is proved that: - the hyperspace of subcontinua of the Long Arc has the trivial Whitney levels only; - the hyperspace of subcontinua of the Lexicographic Square has non-trivial Whitney levels but admit no Whitney maps.