What happens when every dense set is open?

Question

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.

Answer 1

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.