Uniformly discrete hit-and-miss hypertopology. A missing link in hypertopologies

Abstract

[EN] Recently it was shown that the lower Hausdorff metric (uniform) topology is generated by families of uniformly discrete sets as hit sets. This result leads to a new hypertopology which is the join of the above topology and the upper Vietoris topology. This uniformly discrete hit-and-miss hypertopology is coarser than the locally finite hypertopology and finer than both Hausdorff metric (uniform) topology and Vietoris topology. In this paper this new hypertopology is studied. Here is a Hasse diagram in which each arrow goes from a coarser topology to a finer one and equality follows UC or TB as indicated. The diagram clearly shows that the new (underlined) topology provides the missing link.