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Let $X$ be a compact Hausdorff space and let $\{U_n\}$ be a countable collection of subsets that are open and dense in $X$. Show that the intersection $$\bigcap\limits_{n=1}^\infty U_n$$ is dense.

I tried to show that the closure of this intersection is equal to the intersection of the closures of each set, but I'm not getting anywhere.

Any help would be greatly appreciated.

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3  
This makes two questions in the last hour for which the answer is "Actually, this is the Baire Category Theorem..." Coincidence? –  Pete L. Clark Dec 6 '11 at 4:13
    
Could "Student" be Baire himself looking for approval? –  ae0709 Dec 6 '11 at 5:20
    
@aengle: ...and here I always thought that Student's actual surname was "Gosset"... :D –  J. M. Dec 6 '11 at 9:55

1 Answer 1

What you are looking for is Baire's theorem. (The proof on Wikipedia is for a complete metric space, but the proof is similar for a compact Hausdorff space.)

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