On metric boundedness structures
In [J. Math. Pures Appl., IX. Sér. 28, 287-320 (1949; Zbl 0041.31602)] \textit{S.-T. Hu} characterized, given a metrizable space \(X\), those families \(\mathcal S\) of subsets of \(X\) for which a compatible metric \(d\) on \(X\) can be found such that \(\mathcal S\) consists of exactly the \(d\)-bounded subsets of \(X\) --- they are the proper ideals on \(X\) with countable generating sets that contain all singletons and satisfy: if \(S\in\mathcal S\) then there is \(T\in\mathcal S\) such that \(\operatorname{cl}S\subseteq\operatorname{int}T\). The author concentrates on the dual filters of such ideals; such a filter is described completely by specifying a sequence \(\langle F_n\rangle_n\) of closed sets such that \(\bigcap_nF_n=\emptyset\) and \(F_{n+1}\subseteq\operatorname{int}F_n\) for all \(n\) --- the author calls this a metric mode of convergence to infinity. A host of such sequences can be obtained as follows. Generalizing the author's construction a bit we consider a discrete family \(\mathcal O\) of open sets in the space under consideration and choose in each \(O\in\mathcal O\) a sequence \(\bigl<U(O,n)\bigr>_n\) of open sets such that \(\operatorname{cl}U(O,n+1)\subseteq U(O,n)\) for all \(n\). Whenever \(\langle O_m\rangle_m\) is a sequence in \(\mathcal O\) one may put \(F_m=\bigcup_{n\geq m}\operatorname{cl}U(O_n,n)\) for every \(m\) to get a sequence as desired. Because one can always find \(|\mathcal O|^\omega\) sequences in \(\mathcal O\) with pairwise disjoint ranges one can thus construct, in a noncompact metrizable space \(X\) of weight \(\kappa\), at least \(\kappa^\omega\) distinct metric boundedness structures on \(X\). This improves the author's `uncountably many'. Using these structures the author's methods provide at least \(\kappa^\omega\) many different Wijsman (and also Attouch-Weis) topologies on the hyperspace of closed subsets of \(X\).
scientific article; zbMATH DE number 1418151