Characterizations of strongly countable-dimensional and locally finite- dimensional spaces
The author characterizes metrizable strongly countable-dimensional and locally finite-dimensional spaces in terms of three properties: (i) partitions between members of a sequence of pairs of disjoint closed sets, (ii) frontiers of members of a \(\sigma\)-discrete base, and (iii) a closed mapping from a zero-dimensional space. The results parallel characterizations of countable-dimensional metrizable spaces given by Nagata, Nagami, and Roberts. A family \(\{A_ s\): \(s\in S\}\) of subsets of \(X\) is closedly point- finite (openly point-finite) if \(X\) can be represented as the union of a sequence \(X_ 1,X_ 2,\dots\) of closed (open) subsets such that the order of \(\{A_ s\cap X_ n\): \(s\in S\}\) is finite for each \(n=1,2,\dots\;\). Then one of the three characterizations of a strongly countable-dimensional metrizable space is given by the existence of a \(\sigma\)-discrete base for which the family of frontiers is closedly point-finite. The main theorem also contains three analogous characterizations for the locally finite-dimensional case.
scientific article; zbMATH DE number 96241
scientific article