Why is it that if a space is not Hausdorff and you take the intersection of nested compact subspaces the intersection could be empty? Could you give me an example?
Intersection of nested compact subspaces in non-Hausdorff space
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$\begingroup$
general-topology
compactness
1 Answers
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Take $X$ to be any set with the trivial topology (only the empty set and $X$ are open). Every subset of $X$ is compact. Now take any chain of subsets whose intersection is empty.
For example, $\mathbb N$ with the trivial topology and intersect $A_k = \{n\in\mathbb N\mid n>k\}$.
This idea transfer to any space that has the property that every subspace is compact (this is known as a Noetherian space). For example co-finite topologies have this property.
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0@upaudel: I don't know. You should ask this as a new question perhaps. – 2013-04-21