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