Spaces $X$ in which every subset is either open or closed, and only $\varnothing$ and $X$ are clopen Let $(X, \tau)$ be a topological space. Then $X, \varnothing \in \tau$ and are both clopen.
But I wonder if it is possible to construct a topological space $X$ in which all subsets are either open or closed, but $X$ and $\varnothing$ are the only clopen subsets.

Pick any $X$ and $a \in X$.
Define $Y \subset X$ to be open if and only i f$X= \emptyset$ or $a \in Y$. It follows that a set $Y$ is closed if and only if $Y=X$ or $a \notin Y$.
It is easy to show that this is a topology which has the required properties.
It is likely that there is no separable topology with these properties.
