The relationship between the Vietoris topology and the Hausdorff quasi-uniformity
One early result in the study of hyperspace quasi-uniformities is that the Hausdorff quasi-uniformity of a Pervin quasi-uniformity of a topological space \(X\) induces the Vietoris topology on the family of nonempty subsets of \(X\), see [\textit{N. Levine} and \textit{W. J. Stager jun.}, Math. J. Okayama Univ. 15, 101-106 (1972; Zbl 0246.54033)]. The authors extend this result to the other members of the Pervin quasi-proximity class: If \({\mathcal U}\) belongs to the Pervin quasi-proximity class of a topological space \(X\), the Hausdorff quasi-uniformity of \({\mathcal U}\) induces the Vietoris topology on the family of nonempty subsets of \(X\) if and only if \({\mathcal U}^{-1}\) is hereditarily precompact. (As the authors observe, it follows that the Hausdorff quasi-uniformity is compatible with the Vietoris topology for both the Pervin and well-monotone quasi-uniformities, since the inverse of these quasi-uniformities are hereditarily precompact.) The authors also consider the Vietoris topology on \(K_0(X)\), the nonempty compact subsets of \(X\), and on \({\mathcal C}{\mathcal L}_0(X)\), the nonempty closed subsets of \(X\). They establish that the Hausdorff quasi-uniformity of a quasi-uniform space \((X,{\mathcal U})\) induces the Vietoris topology on \(K_0(X)\) if and only if \({\mathcal U}^{-1}|K\) is precompact for each nonempty compact set \(K\) of \((X,{\mathcal I}({\mathcal U}))\). For a \(T_1\) quasi-uniform space \((X,{\mathcal U})\), they show that the Hausdorff quasi-uniformity induces the Vietoris topology on \({\mathcal C}{\mathcal L}_0(X)\) if and only if \({\mathcal U}\) is equinormal and \({\mathcal U}^{-1}\) is hereditarily precompact. Reviewer's remark: The authors' Lemma 1 can be used to show that the quasi-uniformities that satisfy the property QPC are exactly the quasi-uniformities that belong to the Pervin quasi-proximity class; the review incorporates this observation.
scientific article; zbMATH DE number 1879103
scientific article