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==Topological Statement==
Let <math>S</math> be a Hausdorff topological space. A decreasing nested sequence of non-empty compact subsets of <math>S</math> has a non-empty intersection. In other words, supposing (''C''<sub>''k''</sub>) is a sequence of non-empty, compact subsets of <math>S</math> satisfying
:<math>C_0 \supseteq C_1 \supseteq \cdots C_k \supseteq C_{k+1} \cdots, </math>
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