Quasi-uniform hyperspaces of compact subsets
Suppose \((X,{\mathcal U})\) is a quasi-uniform space, \({\mathcal T}_V\) denotes the Vietoris topology and \({\mathcal U}^*\) denotes the Hausdorff-Bourbaki quasi-uniformity on \({\mathcal K}(X)\) which is the space of all non-empty compact subsets of \(X\). It is well known that if \({\mathcal U}\) is a uniformity, then \({\mathcal T}_V\) is compatible with \({\mathcal U}^*\) and \({\mathcal U}^*\) is complete if and only if \({\mathcal U}\) is complete. In this paper the authors show that \({\mathcal T}_V \subset {\mathcal T}({\mathcal U}^*)\) and investigate when they are equal. For this purpose they introduce the concepts ``compactly symmetric'' and ``small-set symmetric'' which strengthen the concept of ``local symmetry''. Main results: (1) If \((X,{\mathcal U})\) is compactly symmetric or small-set symmetric, then \({\mathcal T}_V\) is compatible with \({\mathcal U}^*\). (2) If \((X,{\mathcal U})\) is compactly symmetric, then \({\mathcal U}^*\) is complete if and only if \({\mathcal U}\) is complete. The result is not true for small-set symmetric spaces.
scientific article; zbMATH DE number 1307962
scientific article