Cantor's intersection theorem: Difference between revisions

Revision as of 02:32, 4 February 2019

Revision as of 07:35, 13 October 2019

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

''Note'': We may ruleleave out the closedclosedness condition providedin thatsituations where every compact subset of <math>S</math> is closed, for example when <math>S</math> is [[Hausdorff space|Hausdorff]].

=== Proof ===

Assume, by way of contradiction, that <math>\bigcap C_n=\emptyset</math>. For each n, let <math>U_n=C_0\setminus C_n</math>. Since <math>\bigcup U_n=C_0\setminus\bigcap C_n</math> and <math>\bigcap C_n=\emptyset</math>, we have <math>\bigcup U_n=C_0</math>.