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Hi,

I have a (probably stupid) question about the Baire Category Theorem. I am looking at the statement that says that in a complete metric space, the intersection of countable many dense open sets is dense in the metric space.

Assume that we have the countable collection of dense open sets ## \{U_n\} ## in a complete metric space ##X##, and let ##x \in X, \epsilon>0##. Since ##U_1## is dense in ##X##, there is ##y_1\in U_1## with ##d(x,y_1)<\epsilon##. Also, as ##U_1## is open, there is ##r_1>0## with ##B(y_1;r_1)\subset U_1##.

Now my question is why we can arrange that the closure will be contained in each of them? I think intuitively it sounds correct, but I didn't succeed to prove it rigorously. Can you please help me here?

I have a (probably stupid) question about the Baire Category Theorem. I am looking at the statement that says that in a complete metric space, the intersection of countable many dense open sets is dense in the metric space.

Assume that we have the countable collection of dense open sets ## \{U_n\} ## in a complete metric space ##X##, and let ##x \in X, \epsilon>0##. Since ##U_1## is dense in ##X##, there is ##y_1\in U_1## with ##d(x,y_1)<\epsilon##. Also, as ##U_1## is open, there is ##r_1>0## with ##B(y_1;r_1)\subset U_1##.

**Then, we can arrange ##r_1<1## such that ##\overline{B(y_1;r_1)} \subset U_1\cap B(x;\epsilon) ##.**Now my question is why we can arrange that the closure will be contained in each of them? I think intuitively it sounds correct, but I didn't succeed to prove it rigorously. Can you please help me here?

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