Metric Spaces, Nested Closed Sets

Nested Closed Sets

Let a descending chain of closed sets in a complete metric space have diameters that approach 0. We will show that they intersect in a single point.

Select any point from each set and construct a Cauchy sequence, which implies a limit point p. Suppose p is not in one of the closed sets C. Start the cauchy sequence at C; it is still cauchy with the same limit. The sequence lies entirely in C, and p is its limit point, which puts p in the closure of C. Thus p belongs to C. This holds for every closed set in the chain, hence p is in the intersection.

Any other point q will be a certain distance from p, and will not be contained in the closed sets with lesser diameters. The intersection is precisely p.

This result does not hold for open sets, as illustrated by the intervals (0, 1/n) in R¹. Similarly, closed sets in a non-complete metric space may close in on a missing point, as when shrinking rational intervals approach the square root of 2.