nested sphere theorem

In a metric space $X$ , let $\overline{B}_{r}(x)$ be the closed ball centered at $x\in X$ with radius $r>0$ .

Theorem 1 (Nested sphere theorem [KF]).

A metric space $X$ is complete if and only if every sequence $\{\overline{B}_{r_{n}}(x_{n})\}_{n}$ such that $\overline{B}_{r_{i+1}}(x_{i+1})\subseteq\overline{B}_{r_{i}}(x_{i})$ and $r_{n}\to 0$ when $n\to\infty$ has a nonempty intersection (i.e. $\bigcap_{n=1}^{\infty}\overline{B}_{r_{n}}(x_{n})\neq\emptyset$ ).

References

KF Kolmogorov, A.N. & Fomin, S.V.: Introductory Real Analysis, Translated & Edited by Richard A. Silverman. Dover Publications, Inc. New York, 1970.

Title	nested sphere theorem
Canonical name	NestedSphereTheorem
Date of creation	2013-03-22 14:57:12
Last modified on	2013-03-22 14:57:12
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	5
Author	Daume (40)
Entry type	Theorem
Classification	msc 54E35
Classification	msc 54E50