Non-trivial topology where only open sets are closed For example, on $\mathbb{R}$ there exists trivial topology which contains only $\mathbb{R}$ and $\emptyset$ and in that topology all open sets are closed and all closed sets are open.
Question. Does there exist non-trivial topology on $\mathbb{R}$ for which all open sets are closed and all closed sets are open? Further, given some general set $X$ whose number of elements is finite, could we always construct non-trivial topology where all open sets are closed and all closed sets are open? What if $X$ has a non-finite number of elements?
I hope my question is not meaningless.
Thank you for any help.

You can manufacture examples like this pretty easily.  Let $A$ be any subset of $\mathbb{R}$.  Put a topology on $\mathbb{R}$ with the following open sets:
$$
\varnothing, A, \mathbb{R}-A, \mathbb{R}.
$$
You can easily check that this always gives a topology, and a subset of $\mathbb{R}$ is open if and only if it is closed.   This construction generalizes to any set $X$:  there's nothing special about $\mathbb{R}$ here. 
