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==Topological Statement==
Let <math>S</math> be a [[Topological Space|topological space]]. A decreasing nested sequence of non-empty compact, closed subsets of <math>S</math> has a non-empty intersection. In other words, supposing (''C''<sub>''k''</sub>) is a sequence of non-empty
:<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>
''Note'': We may rule out the closed condition provided that every compact subset of <math>S</math> is closed, for example when <math>S</math> is 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>.
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