Hello everyone! thanks for reading!

I need to prove the following...

Let X be a topological space with a countable basis, that satisfies the "Uniqueness of limit" axiom, meaning, that every sequence converges only into one limit.

Then X is hausdorff.

In class, we have studied of spereration axioms, marked as T0, T1, T2, etc...

A T1 space is one that for every x,y, x<>y (x isn't y), you can find an open set of x, not intersecting y, and vice versa.

A T2 space is an hausdorff space - for every x,y, x<>y, you can find two DIJOINT neighborhoods of x, y.

I have shown that the uniqueness of limit means that X is T1. I cannot, however, find a way to use the countability of the basis to form two disjoint open sets around x and y....

Any hints/help/answers greatly appreciated

Tomer.