On nonempty intersection properties in metric spaces

Abstract

The classical Cantor’s intersection theorem states that in a complete metric space X, intersection of every decreasing sequence of nonempty closed bounded subsets, with diameter approaches zero, has exactly one point. In this article, we deal with decreasing sequences {Kn} of nonempty closed bounded subsets of a metric space X, for which the Hausdorff distance H(Kn, Kn+1) tends to 0, as well as for which the excess of Kn over X \ Kn tends to 0. We achieve nonempty intersection properties in metric spaces. The obtained results also provide partial generalizations of Cantor’s theorem.