nested sphere theorem

In a metric space X, let B¯r(x) be the closed ball centered at xX with radius r>0.

Theorem 1 (Nested sphere theorem [KF]).

A metric space X is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if and only if every sequencePlanetmathPlanetmath {B¯rn(xn)}n such that B¯ri+1(xi+1)B¯ri(xi) and rn0 when n has a nonempty intersectionMathworldPlanetmathPlanetmath (i.e. n=1B¯rn(xn)).


  • 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