Cantor's intersection theorem: Difference between revisions

Revision as of 02:32, 4 February 2019

Revision as of 02:32, 4 February 2019

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== Variant in complete metric spaces ==

In a [[complete metric space]], the following variant of Cantor's intersection theorem holds. Suppose that ''X'' is a non-empty complete metric space, and ''C''_''n'' is a sequence of non-empty closed nested subsets of ''X'' whose [[diameter]]s tend to zero:

:<math>\lim_{n\to\infty} \operatorname{diam}(C_n) = 0</math>

where diam(''C''_''n'') is defined by