Universal elements for families of separable metrizable spaces
The authors describe a very general way of constructing universal elements in families of separable metric spaces. The rough idea is to represent each space by a pair \((S,D)\), where \(S\) is a subset of the Cantor set \(C\) and \(D\) is an upper-semicontinuous decomposition of \(S\) with compact fibers. The number of these pairs is reduced to `at most \(\mathfrak c\)' by amalgamating equivalent pairs into one, where the equivalence of \((S,D)\) and \((S',D')\) seems to mean that \(S\) and \(S'\) have the same closure and that \(D\cup D'\) is upper-semicontinuous on \(S\cup S'\). The set of \(\mathcal E\) of these amalgamations is topologized so as to be separable metric. Their decompositions induce an upper-semicontinuous decomposition of the subspace \[ \bigcup\Bigl\{\bigl\{(S,D)\bigr\}\times S:(S,D)\in{\mathcal E}\Bigr\} \] of the product \(C\times\mathcal E\); the corresponding quotient space \(Q\) contains a copy of every element of the family. To make \(Q\) an element of the family, for example if the family is defined by some dimensional condition, the authors have to develop a huge amount of notation for bookkeeping.
scientific article; zbMATH DE number 1103438
scientific article