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Compactness for a class of hit–and–miss hyperspaces. (English) Zbl 1194.54033
Summary: In the study of some kind of generalized Vietoris-type topologies for the hyperspace of all nonempty closed subsets of a topological space \((X,\tau)\), namely the so called \(\Delta\)-hit-and-miss-topologies with \(\Delta\subset Cl(X)\) (or \(\Delta\)-topologies), which was initiated by the second author in 1965 [Arch. Math. 16, 197-199 (1965; Zbl 0127.13004)], it is obvious, that the non-compactness of such a hyperspace often depends on the non-compactness even in the lower-semifinite topology (induced by the “hit-sets”), which is contained in all hypertopologies of this type. Otherwise, compactness for these topologies is easily obtained from the compactness of \((X,\tau)\) by well-known theorems, if the “miss-sets” are induced either by compact or closed subsets. To obtain a similar result for topologies with “miss-sets” generated by subsets with a property which generalizes both, closedness and compactness especially in the non-Hausdorff case, we use consequently a quite set-theoretical lemma, stated at the beginning.
MSC:
54D30 Compactness
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