I've looked on the web and haven't found a simple example.



Example of a weak Hausdorff space that is not Hausdorff?
The one-point compactification of $\mathbb{Q}$ has the property that every compact subset is closed. So it is certainly a weak Hausdorff space. But it isn't Hausdorff, as $\mathbb{Q}$ isn't locally compact.


**Addendum**


Another example is the [cocountable topology](http://en.wikipedia.org/wiki/Cocountable_topology) on an uncountable set. No two points have disjoint neighbourhoods, and the only compact subsets are the finite subsets.


