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I came across a property in a textbook that caught my eye. The property is: If $X$ is a complete metric space, then the intersection of any two dense $G_{\delta}$-subsets of $X$ is dense in $X$.

This property seems simple, but I am having trouble figuring out the proof. Can anyone help me out on this one?

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up vote 6 down vote accepted

This is an immediate consequence of the Baire category theorem and the fact that if $G$ is a $G_\delta$ in $X$, then $G=\bigcap_{n\in N}U_n$ for some open sets $U_n$ in $X$, and if in addition $G$ is dense in $X$, then so is each of the sets $U_n$.

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