On Hausdorff Metric Spaces

Abstract

An {\it expansive mapping of Lipschitz type} is introduced. A map, induced by a given map $T$ between two metric spaces $X$ and $Y$, from the power set of $X$ to the power set of $Y$ is considered. It is proved that the induced map preserves continuity, Lipschitz continuity and expansiveness of Lipschitz type. A nonempty intersection property in a metric space is achieved which also provides a partial generalization of the classical Cantor's Intersection Theorem. Using this nonempty intersection property and the considered induced map, it is shown that the converse of Henrikson's result (i.e. a Hausdorff metric space is complete if its underlying space is complete) also holds.Comment: 14 page