It is known that there are non-Hausdorff spaces which admit unique limits for all convergent sequence (see [here](http://www.sciencedirect.com/science/article/pii/0166864193901476)) and it is also known that unique limits for nets implies Hausdorff.


What I am wondering is, if there is a (somehow weak) condition which one should add to "unique limits of sequences" to obtain a Hausdorff space. Would, for example, some countability help?


Somehow in the same direction: What is the central property which is needed for a space such that it can be non-Hausdorff but has unique sequence limits? Is there a whole class of non-Hausdorff spaces which admit unique limits for convergent sequence?



Unique limits of sequences plus what implies Hausdorff?
[First countable](http://en.wikipedia.org/wiki/First-countable_space) is enough. Let $x\neq y$ be two points in your space that cannot be separated by neighborhoods. Let $O\_1,O\_2,\ldots$ form a neighborhood base of $x$ and let $U\_1,U\_2,\ldots$ form a neighborhood base for $y$. Choose a sequence $(z\_n)$ such that $z\_n\in O\_n\cap U\_n$ for all $n$. Now $(z\_n)$ converges to both $x$ and $y$.


