What characterizes topological spaces where every open set is closed? Motivated by the valuation topology on a discrete valuation ring, which has the above property, I want to know if there's some (subjectively, probably) nicer criterion for when a space has every open set closed. A preliminary guess (justified again by discrete valuation rings) is that a totally disconnected space has the above property; but this seems qualitatively too simple to actively characterize such a space, and certainly there's no apparent proof that totally disconnected implies this clopenness property. (Considering I'm mostly interested in this in the setting of topological rings, I'd be interested in the class of rings that satisfy this property, too, and thus the abstract algebra tag.)
 A: The topology on a valuation ring does not have this property.  (E.g. points are closed, but not open, hence there complements are open, but not closed.)
In such a space, the closure of any point is the closure of each of its points.  Thus the property of having the same closure is an equivalence relation on points, and if we take the quotient by this equivalence relation, we get a set with the discrete topology.
In summary, there is a surjection $X \to Y$, such that the topology on $X$ is just the pull-back of the discrete topology on $Y$.
A: The name for such a property is locally indiscrete, which may help you search the literature.
An excerpt from: Dontchev, J. (1998). Survey on preopen sets. arXiv preprint math/9810177.

Recall that a space $(X,\tau)$ is called locally indiscrete if every open subset of $X$ is closed.
$\bf{Theorem\; 3.3}$ For a topological space $(X,\tau)$, the following conditions are equivalent:
$(1)$ $X$ is locally indiscrete.
$(2)$ Every subset of $X$ is preopen.
$(3)$ Every singleton in $X$ is preopen.
$(4)$ Every closed subset of $X$ is preopen.

A: It seems the following. 
Let $X$ be a topologcal space such that each open subset of $X$ is closed. Then each closed subset of $X$ is open and $X$ is a disjoint sum of its antidiscrete components. For a point $x\in X$ its component $X_x$ is eaual to the intersection of all closed (or open) sets containing the point $x$.
