Cantor's intersection theorem, also known as Cantor's nested intervals theorem, refers to a pair of closely related results in general topology and real analysis, both named after Georg Cantor. The theorems concern what happens when an infinite sequence of nonempty compact sets is nested inside one another, each contained in the one before it, and they establish that under these conditions the intersection of the entire sequence is guaranteed to be nonempty. This guarantee depends specifically on the sets being compact and nested in a decreasing chain, and it is a foundational tool used throughout analysis and topology wherever an infinite nested sequence of sets needs to be shown to have a common point.
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StatementLet S be a topological space. A decreasing nested sequence of non-empty compact, closed subsets of S has a non-empty intersection. 1 Classification
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Missing in-branch edge found while every sibling theorem entity in this pass already carried both in-category and in-branch; the entity already carries in-category.
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Source Cantor's Intersection Theorem (Wikipedia)
Sources
1. Cantor's Intersection Theorem (Wikipedia)
Wikimedia FoundationLead section
Cantor's intersection theorem, also called Cantor's nested intervals theorem, refers to two closely related theorems in general topology and real analysis, named after Georg Cantor, about intersections of decreasing nested sequences of non-empty compact sets.
In Group: General Topology, Lead paragraph
Cantor's intersection theorem, also called Cantor's nested intervals theorem, refers to two closely related theorems in general topology and real analysis, named after Georg Cantor, about intersections of decreasing nested sequences of non-empty compact sets.
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