Nested nonempty closed subsets of a compact space have nonempty intersection.Suggest a slogan
Let \(X\) be a compact space and let \(C_1 \supseteq C_2 \supseteq \cdots\) be a decreasing sequence of nonempty closed subsets of \(X\). Then \(\Intersect_n C_n \neq \emptyset\) [@Mun00].
If \(X\) is Hausdorff, it suffices that each \(C_n\) be nonempty and compact, since compact subsets of Hausdorff spaces are closed [@Mun00]. Without closedness the conclusion fails: in an infinite set \(\NN\) with the indiscrete topology every subset is compact, and \(C_n = \ts{n, n+1, \ldots}\) is a decreasing sequence of nonempty compact sets with empty intersection.