Assume that the only dense subset is $X$ itself. What can you say something about the topology, that is, the family of open sets? Let $(X, d)$ be a metric space. Assume that the only dense subset is $X$ itself. What can you say something about the topology, that is, the family of open sets?
My Try: Since the only dense subset of $X$ is $X$ itself, so every subset of $X$ is a closed set.

One thought: Every point is isolated. For if $x$ is a limit point of $X$, then the closure of $X-x$ is $X$. We assume this cannot happen, so $x$ must be isolated; hence the singleton of the arbitrary point $x$ is open. 
