  We study two properties for subsets of a metric space. One of them is
generalization of chainability, finite chainability, and Menger convexity for
metric spaces; while the other is a generalization of compactness. We explore
the basic results related to these two properties. Further, in the perspective
of these properties, we explore relations among the Hausdorff, Vietoris, and
locally finite hypertopologies.
