Is there a generalization of the nested interval theorem in $\mathbb{R}^n$?

Question

I'm familiar with the nested interval theorem on the real line. But is there a generalization of such a theorem in literature?

Answer 1

There is a generalization to metric and Hausdorff topological spaces. Any nested sequence of non-empty compact subsets has non-empty intersection. For metric spaces, if the diameter approaches zero, the intersection is one point. So for $\mathbb{R}^n$, any nested sequence of (closed) boxes or closed balls has non-empty intersection.

Actually, more is true (finite intersection property): If a family of compact sets has the property that the intersection of any finite number of them is non-empty, then their intersection is non-empty. Obviously nested families enjoy the finite intersection property.

Answer 2

More generally, suppose we have compact sets $X = U_0\supset U_1\supset U_2 \supset \cdots$ in some arbitrary compact, Hausdorff space $X$. If $\bigcap U_i$ is empty, then the sets $V_i = X\setminus U_i$ form an open cover of $X$ and must therefore admit a finite subcover; that is, some $U_i$ is empty.

(If you're not familiar with compact spaces, the key points are that bounded, closed subspsaces of $\mathbb{R}^N$ (e.g., closed boxes) satisfy the hypothesis above; and for $X$ compact, by definition any open sets $U_i\subset X$ with $\bigcup U_i = X$ have $\bigcup_{i\in S} U_i = X$ for a finite set $S$.)