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

Abstract

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.

Keywords

• Hypertopology
• Hausdorff metric
• Vietoris topology
• Proximal topology
• Proximal locally finite
• Hit-and-miss hypertopology
• ε-discrete subset
• Uniformly discrete
• Uniformly isolated
• locally finite family