Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

1 Answer 1

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 $U_n$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.