Hausdorff Spaces Subsets open? 'Given a Hausdorff space $X$ with finitely many elements, show all subsets of $X$ are open in $X$.'
I let $U$ be an arbitrary subset of $X$
Since every subset of a Hausdorff space is a Hausdorff space, then $U$ is also Hausdorff. 
This means that any two points in $U$ $x \ne y$ have disjoint neighbourhoods.
Does this imply every point of $U$ has a neighbourhood lying in $U$ hence prove open? This seems trivial. Where does finitely many elements come into it?

We only need that singletons are closed (because then all finite sets are closed, and so all sets are open)...
