# Thread: If no proper subset of X is dense..

1. ## If no proper subset of X is dense..

If no proper subset of the topological space X is dense, is the topology necessarily discreate?

2. ## Re: If no proper subset of X is dense..

Take $x\in X$ and let $A:=\complement\{x\}$. A is not dense, and $\overline{A}=\overline{\complement\{x\}}=\overset{ \circ}{\{x\}}^c$ hence $\overset{\circ}{\{x\}}\neq \emptyset$ and $\{x\}$ is open.