A short note on hit- and miss-hyperspaces

 René Bartsch, Harry Poppe

  In the study of some kind of generalized Vietoris--type topologies for the hyperspace of all nonempty closed subsets of a topological space (X,t), namely the so called D hit-and-miss-topologies with D Í Cl(X) (or D-topologies), which was initiated by the second author in 1965, 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,t) 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  called “ weak relative completeness”, which generalizes both, closedness and compactness especially in the non-Hausdorff case, we use consequently a quite set-theoretical lemma, stated at the beginning. Weak relative completeness is characterized by convergence and by open coverings.