On Matheron theorem for non-locally compact metric spaces (Q1573078)

The authors consider for the space of all closed subsets of a metric space a hyperspace topology which is (within the last two decades) known as Fell topology \(\tau_F\). The authors' source is [\textit{G. Matheron}, Random sets and integral geometry (1975; Zbl 0321.60009)]. Probably, Matheron rediscovered \(\tau_F\). \textit{J. M. G. Fell} defined and studied this topology in [Proc. Am. Math. Soc. 13, 472-476 (1962; Zbl 0106.15801)]. For a topological space \((X,\tau)\), let \(2^X\) denote the family of closed subsets of \(X\), including the empty set \(\emptyset\). Fell proved, that \((2^X,\tau_F)\) is compact and Hausdorff if \(X\) is locally compact. The reviewer proved in [Fundam. Math. 59, 159-169 (1966; Zbl 0139.40404)]: Let \((X,\tau)\) be a \(T_1\)-space, \(\text{Cl}(X):=2^X\smallsetminus\{\emptyset\}\), \(\alpha:=\{B\subseteq X\mid B\) is closed and compact\}, \(B^+:=\{A\in \text{Cl}(X)\mid A\cap B=\emptyset\}\) (miss-set) and \(G^-:=\{A\in \text{Cl}(X)\mid A\cap G\neq\emptyset\}\) (hit-set). Let \(\tau_\alpha\) be the topology for \(\text{Cl}(X)\) which is generated by the subbase consisting of all \(B^+\), \(B\in \alpha\) and \(G^-\), \(G\) open. Then the following propositions are equivalent: (1) \((\text{Cl}(X),\tau_\alpha)\) is Hausdorff; (2) each neighbourhood of each point \(x\in X\) contains a neighbourhood of \(x\), which belongs to \(\alpha\); (3) \((\text{Cl}(X),\tau_{\alpha})\) is regular (where regular implies Hausdorff). Of course, if \((X,\tau)\) is Hausdorff, we have \(\tau_\alpha=\tau_F\). [For instance, see also \textit{G. Beer}, Set-Valued Anal. 1, No. 1, 69-80 (1993; Zbl 0810.54010), \textit{L. Zsilinszky}, Rend. Circ. Mat. Palermo, II. Ser. 49, No. 2, 371-380 (2000; Zbl 0980.54006).] Matheron proved for a locally compact (Hausdorff and second countable) space \((X,\tau)\) (again), that \((2^X,\tau_F)\) is Hausdorff and compact and that \((2^X,\tau_F)\) is second countable. The authors prove for a Polish space \(X\) (complete, separable, metric) again, that \((2^X,\tau_F)\) is compact, that \((2^X,\tau_F)\) is separable, and that \((2^X,\tau_F)\) cannot be Hausdorff if \(X\) has a point where the neighbourhood filter has no base consisting of compact neighbourhoods. Altogether, one must say, that the results of the authors are not new.