Cantor's intersection theorem: Difference between revisions

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In [[real analysis]], a branch of mathematics, '''Cantor's intersection theorem''', named after [[Georg Cantor]], is a theorem related to [[compact set]]s in '''R''', the set of [[real number]]s. It states that a decreasing nested [[sequence]] of non-empty, [[closed set|closed]] and [[bounded set|bounded]] subsets of '''R''' has nonempty intersection. In other words, supposing {C_''k''} is a sequence of non-empty, closed and bounded sets satisfying

:<math>C_0 \supseteq C_1 \supseteq \cdots C_k \supseteq C_{k+1} \cdots </math>

it follows that

:<math>\left(\bigcap_{k} C_k\right) \neq \emptyset</math>.

The result is typically used as a lemma in proving the [[Heine-Borel theorem]], which states that sets of real numbers are compact if and only if they are closed and bounded. Conversely, if the Heine-Borel theorem is known, then it can be restated as: a decreasing nested sequence of non-empty, compact subsets of '''R''' has nonempty intersection.

The result is typically used as a lemma in proving the [[Heine-&ndash;Borel theorem]], which states that sets of real numbers are compact if and only if they are closed and bounded. Conversely, if the Heine-&ndash;Borel theorem is known, then it can be restated as: a decreasing nested sequence of non-empty, compact subsets of '''R''' has nonempty intersection.

As an example, if C_''k'' = [0, 1/''k''], the intersection over {C_''k''} is {0}. On the other hand, both the sequence of open bounded sets C_''k'' = (0, 1/''k'') and the sequence of unbounded closed sets C_''k'' = [k, ∞) have empty intersection. All these sequences are properly nested.

As an example, if ''C''_''k'' = &nbsp;[0, &nbsp;1/''k''], the intersection over {''C''_''k''} is &nbsp;{0}. On the other hand, both the sequence of open bounded sets ''C''_''k'' = &nbsp;(0, &nbsp;1/''k'') and the sequence of unbounded closed sets ''C''_''k'' = &nbsp;[''k'', &nbsp;∞) have empty intersection. All these sequences are properly nested.

The theorem generalizes to '''R'''^''n'', the set of ''n''-element vectors of real numbers, but does not generalize to arbitrary [[metric space]]s. For example, in the space of [[rational number]]s, the sets <math>C_k = [\sqrt{2}, \sqrt{2}+1/k] = (\sqrt{2}, \sqrt{2}+1/k)</math> are closed and bounded, but their intersection is empty.

: <math>C_k = [\sqrt{2}, \sqrt{2}+1/k] = (\sqrt{2}, \sqrt{2}+1/k)</math>

are closed and bounded, but their intersection is empty.

The theorem generalizes to '''R'''^''n'', the set of ''n''-element vectors of real numbers, but does not generalize to arbitrary [[metric space]]s. For example, in the space of [[rational number]]s, the sets <math>C_k = [\sqrt{2}, \sqrt{2}+1/k] = (\sqrt{2}, \sqrt{2}+1/k)</math> are closed and bounded, but their intersection is empty.

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.

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== Proof ==

Consider the sequence (''a''_''k'') where ''a''_''k'' is the [[infimum]] over the non-empty ''C''_''k''. Because ''C''_''k'' is closed, ''a''_''k'' belongs to ''C''_''k''; because the sets are decreasing nested, the sequence is monotonic increasing. Because it is also bounded (being contained in the bounded set ''C''₁), it must converge to some limit &nbsp;''L''. Choose any ''j'' &nbsp;&ge; &nbsp;1; the subsequence of (''a''_''k'') for ''k'' &nbsp;&ge; &nbsp;''j'' is contained in C_''j'' and converges to &nbsp;''L''. Since ''j'' is closed, L lies in &nbsp;''C''_''j''. Since this is true for all ''j'', ''L'' lies in all ''C''_''j'', and so in their intersection.

== References ==