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i have this trouble, but no idea how to star, so I need some help!

Let X be a Hausdorff space, and let $\{C_\alpha| \alpha \in A \}$ a family of closed subsets of $X$ such that $\bigcap C_\alpha \neq \emptyset$. Let U and open that contains $ \bigcap C_\alpha $. Prove that for each $C_{\alpha0} $ compact exist $C_{\alpha1},C_{\alpha2},..., C_{\alpha n},$ such that $ C_{\alpha1} \bigcap C_{\alpha2} \bigcap ... \bigcap C_{\alpha n} \subset U$?

Thank you!!

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    Are the $C_\alpha$ compact? If not, what are the $C_{\alpha n}$'s exactly? Are they subsets of some $C_\alpha$ or so?2012-11-30
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    It still doesn’t quite make sense. Is $C_{\alpha 0}$ one of the sets in the family, one that just happens to be compact? And what do the sets $C_{\alpha 1},\dots,C_{\alpha n}$ have to do with $C_{\alpha 0}$?2012-11-30
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    $ C_{\alpha 0}$ belongs to the family, and sorry didn´t write well the end: $ C_{\alpha 0} \bigcap C_{\alpha 1} \bigcap ... C_{\alpha n} \subset U$2012-11-30
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    That was my guess, but I wanted to be sure.2012-11-30

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