nested sphere theorem

In a metric space $X$ , let $\\overline{B}_{r}(x)$ be the closedball 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. $\\bigcap_{n=1}^{\\infty}\\overline{B}_{r_{n}}(x_{n})\eq\\emptyset$ ).

References

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

单词	NestedSphereTheorem
释义	nested sphere theorem In a metric space $X$ , let $\\overline{B}_{r}(x)$ be the closedball 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. $\\bigcap_{n=1}^{\\infty}\\overline{B}_{r_{n}}(x_{n})\eq\\emptyset$ ). 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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