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The proof for Baire category theorem in my book assumes that the intersection of dense subsets of a complete metric space is nonempty. Why can this be assumed, why is it true?

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It's not true the way you stated it (think $\Bbb Q$ and the irrationals $\Bbb P$ in $\Bbb R$, e.g.)

But two (or finitely many) open dense sets always intersect, because dense sets intersect all (non-empty) open sets... In any space the finite intersection of finitely many dense open sets is dense and open, Baire strengthens this to countably infinite intersections in some spaces (we lose openness of the intersection, but keep dense-ness).

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  • $\begingroup$ In the title of the question he wrote *open" dense subset, but in the body of the question he didn't write "open"...what is the correct option I can't tell. $\endgroup$ – DonAntonio Feb 19 at 21:08

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