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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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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? – mathmath8128 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

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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The proof is in $\mathbb{R}^k$, but I think the proof is analogous for a compact Haussdorf set.

Let $\{Un\}$ be a countable collection of subsets that are open and dense in X.

Let $O$ be an open set, since $U_1$ is dense, it must intersect $U_1$ in a non-empty open set $O_1$. Let $x_1 \in O_1$ , and choose $r_1$ such that the closed ball $\overline{B_{r_1}(x_1)}$ is contained in $O_1$. Now, because $U_2$ is dense, intersect the ball $B_{r_1}(x_1)$ in a non-empty open set $O_2$. In the same way, let $x_2 \in O_2$ , and choose $r_2$ such that the closed ball $\overline{B_{r_2}(x_2)}$ is contained in $O_2$.

With this process we obtained a nested sequence if non-empty compact sets $\overline{B_1} \supseteq \overline{B_2} \supseteq \overline{B_3} \supseteq ... \supseteq \overline{B_n} \supseteq ... $ . If $ x \in \overline{B_n} $, then $x \in O_n$, for each $n$ and hence $x \in O \cap U|n$. Thus $\bigcap U_n$ intersects each non-empty set $O$ in a least one point, that is precisely, that $\bigcap U_n$ is dense in $X$.

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While this link may answer the question, you should include the essential parts of the answer here. – N. F. Taussig Feb 22 at 15:06
Thank you, I already did. – Astrid A. Olave H. Feb 22 at 21:23

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