Difference between revisions of "Cantor Intersection Theorem"
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(New page: The '''Cantor Intersection Theorem''' states: <blockquote> If '''<math>F_n</math>''' is a collection of closed subspaces of a complete metric space such that <math>F_{n+1} \in F_n</ma...) |
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The '''Cantor Intersection Theorem''' states: | The '''Cantor Intersection Theorem''' states: | ||
<blockquote> | <blockquote> | ||
| − | If '''<math>F_n</math>''' is a collection of closed subspaces of a [[complete metric space]] such that <math>F_{n+1} \in F_n</math> for all <math>n</math> and <math>\lim_{n \to \infty} diameter(F_n) = 0</math>, then '''∩ <math>F_n</math>''' is a single point. | + | If '''<math>F_n</math>''' is a collection of closed subspaces of a [[complete metric space]] such that <math>F_{n+1} \in F_n</math> for all '''<math>n</math>''' and <math>\lim_{n \to \infty} diameter(F_n) = 0</math>, then '''∩ <math>F_n</math>''' is a single point. |
</blockquote> | </blockquote> | ||
[[category: mathematics]] | [[category: mathematics]] | ||
Revision as of 18:04, April 6, 2007
The Cantor Intersection Theorem states:
If is a collection of closed subspaces of a complete metric space such that for all and , then ∩ is a single point.
