Question

Prove or disaprove .The intersection of two dense subsets of a metric space is also dense in it

Answer 1

Consider the metric space $\R$ of real numbers. The sets $\mathbb Q$ of rational numbers and $\R\setminus\mathbb Q$ of irrational numbers are dence subsets in $\R,$ but their intersection $\mathbb Q\cap(\R\setminus\mathbb Q)=\emptyset$ is not dence in $\R.$

We conclude that in general case the intersection of two dense subsets of a metric space is not dense in it.