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==Topological Statement==
'''Theorem.''' ''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 <math>(C_k)_{k \geq 0}</math> is a sequence of non-empty compact, closed subsets of S satisfying''
:<math>C_0 \supset C_1 \supset \cdots \supset C_n \supset C_{n+1} \supset \cdots, </math>
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