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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?

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1 Answer 1

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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Thank you for the help –  vic Dec 3 '12 at 2:36
    
If the space is non-Hausdorff, what can we say about the nested intersection of connected subsets? Is the intersection connected? –  user62775 Apr 21 '13 at 18:57
    
@upaudel: I don't know. You should ask this as a new question perhaps. –  Asaf Karagila Apr 21 '13 at 19:15

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