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 inof '''R''',a thecompact setspace of [[real number]]s<math>S</math>. It states that a decreasing nested [[sequence]] of non-empty, [[closed set|closed]] and [[bounded set|bounded]]compact subsets of '''R'''<math>S</math> 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>

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

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'''a compact space has nonempty intersection.

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.

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== Proof ==

Suppose that <math>\bigcap C_n=\emptyset</math>. Let <math>U_n=X\setminus C_n</math>. Since <math>\bigcup U_n=X\setminus\bigcap C_n</math> and <math>\bigcap C_n=\emptyset</math>, then <math>\bigcup U_n=X</math>.

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;≥&nbsp;1; the subsequence of (''a''_''k'') for ''k''&nbsp;≥&nbsp;''j'' is contained in C_''j'' and converges to&nbsp;''L''. Since ''C''_''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.

Since <math>X</math> is compact and <math>(U_n)</math> is an open cover of it, we can extract a finite cover. Let <math>U_k</math> be the largest set of this cover, then <math>\bigcap C_n=C_k\neq\emptyset</math> by hypothesis.

== References ==