Is the intersection of dense sets dense? Recently I came across the fact that the intersection of two dense sets is a dense set. 
Take $\Bbb{R}$. Clearly, both $\Bbb{Q}$ and $\Bbb{R}\setminus\Bbb{Q}$ are dense in $\Bbb{R}$. However, $\Bbb{Q}\cap (\Bbb{R}\setminus\Bbb{Q})=\emptyset$. How can the $\emptyset$ be dense?

Result: The intersection of two open dense subsets $U,U'\subset X$ of a topological space $X$ is dense.   
To prove this,  it is best to  define "dense"  as: " has non-empty intersection with an arbitrary non empty open subset   $ \emptyset \subsetneq V\subset X$".
So, in our case,  $V\cap U\neq \emptyset $ and then $(V\cap U)\cap U'\neq\emptyset$ also since $V\cap U$ is a non empty open subset of  $ X$ too.
Thus $(V\cap U)\cap U'=V\cap( U\cap U')\neq\emptyset$, proving (by our definition)  that $ U\cap U'$ is dense.
