On Nonempty Intersection Properties in Metric Spaces

TL;DR

This paper explores conditions under which decreasing sequences of closed bounded sets in metric spaces have nonempty intersections, extending Cantor's theorem to broader contexts involving Hausdorff distance and excess measures.

Contribution

It introduces new intersection properties for sequences of closed sets in metric spaces, generalizing Cantor's theorem using Hausdorff distance and excess conditions.

Findings

Nonempty intersections under Hausdorff distance convergence

Generalizations of Cantor's intersection theorem

Partial results for metric space sequences

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 $\{K_n\}$ of nonempty closed bounded subsets of a metric space $X$, for which the Hausdorff distance $H(K_n, K_{n+1})$ tends to $0$, as well as for which the excess of $K_n$ over $X\setminus K_n$ tends to $0$. We achieve nonempty intersection properties in metric spaces. The obtained results also provide partial generalizations of Cantor's theorem.