Cantor's intersection theorem: Difference between revisions

Revision as of 07:55, 9 October 2020

Revision as of 18:55, 2 March 2021

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==Statement for Real Numbers==

The theorem in real analysis draws the same conclusion for [[closed set|closed]] and [[bounded set|bounded]] subsets of the set of [[real number]]s <math>\mathbfmathbb{R}</math>. It states that a decreasing nested sequence <math>(C_k)</math> of non-empty, closed and bounded subsets of <math>\mathbfmathbb{R}</math> has a non-empty intersection.

This version follows from the general topological statement in light of the [[Heine&ndash;Borel theorem]], which states that sets of real numbers are compact if and only if they are closed and bounded. However, it is typically used as a lemma in proving said theorem, and therefore warrants a separate proof.

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A simple corollary of the theorem is that the [[Cantor set]] is nonempty, since it is defined as the intersection of a decreasing nested sequence of sets, each of which is defined as the union of a finite number of closed intervals; hence each of these sets is non-empty, closed, and bounded. In fact, the Cantor set contains uncountably many points.

'''Theorem.''' ''Let'' <math>(C_k)</math> ''be a family of non-empty, closed, and bounded subsets of'' <math>\mathbfmathbb{R}</math> ''satisfying''

:<math>C_0 \supset C_1 \supset \cdots C_n \supset C_{n+1} \cdots. </math>

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''Proof.'' Each nonempty, closed, and bounded subset <math>C_k\subset\mathbfmathbb{R}</math> admits a minimal element <math>x_k</math>. Since for each ''k'', we have

:<math>x_{k+1} \in C_{k+1} \subseteq C_k</math>,