# What happens when every dense set is open?

I am supposed to prove or disprove the following claim:

If in space $$(X, \mathcal{O})$$ every dense set is open, then $$(X, \mathcal{O})$$ is not $$T_2$$-space.

I tried taking arbitrary $$x, y \in X$$ such that $$x \ne y$$. But what now? No one guarantees me that every neighborhood $$U$$ of $$x$$ is dense, so it will intersect every neighborhood $$V$$ of $$y$$. On the other hand, to find a counterexample seems hard, because every finite $$T_2$$-space is discreet. Any help is appreciated.

Consider the discrete topology on a set $$X$$.
• Clearly $$X$$ is Hausdorff.
• The only dense subset of $$X$$ is $$X$$ itself, which is open. So every dense subset of $$X$$ is open.