If no proper subset of the topological space X is dense, is the topology necessarily discreate?
Take $\displaystyle x\in X$ and let $\displaystyle A:=\complement\{x\}$. A is not dense, and $\displaystyle \overline{A}=\overline{\complement\{x\}}=\overset{ \circ}{\{x\}}^c$ hence $\displaystyle \overset{\circ}{\{x\}}\neq \emptyset$ and $\displaystyle \{x\}$ is open.