hausdorff, intersection of all closed sets Can you please help me with this question?

Let's $X$ be a topological space.
  Show that these two following conditions are equivalent :
  
  
*
  
*$X$ is hausdorff
  
*for all $x\in X$ intersection of all closed sets containing the neighborhoods of $x$ it's $\{x\}$.

Thanks a lot!

HINTS: 


*

*If $x$ and $y$ are distinct points in a Hausdorff space, they have disjoint open nbhds $V_x$ and $V_y$, and $X\setminus V_y$ is a closed set containing $V_x$.

*If $F$ is a closed set containing an open nbhd $V$ of $x$, then $V$ and $X\setminus F$ are disjoint open sets.
