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单词 NestedSphereTheorem
释义

nested sphere theorem


In a metric space X, let B¯r(x) be the closedball 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)).

References

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