Is there a "natural" topology on powersets? Let's say a topology T on a set X is natural if the definition of T refers to properties of (or relationships on) the members of X, and artificial otherwise. For example, the order topology on the set of real numbers is natural, while the discrete topology is artificial.
Suppose X is the powerset of some set Y. We know some things about X, such as that some members of X are subsets of other members of X. This defines an order on the members of X in terms of the subset relationship. But the order is not linear so it does not define an order topology on X.
I haven't found a topology on powersets that I would call natural. Is there one?

The question may be  more relevant to your motivation  if you suppose $Y$ is a topological space and you ask for a "reasonable" topology on the power set of $Y$. This is quite tricky  but there are topologies on the family of closed subsets, or of open subsets, see for example 
R. Brown and A.M. Abd-Allah,  ``A compact-open topology on partial
maps  with open domain'', J. London Math Soc. (2) 21 (1980)
480-486. 
The idea is to use the Sierpinski space $S= \{0,1\}$ in which $1$ is open but not closed. Then the set $\mathcal O(Y)$ of open sets of $Y$ is given by a corresponding set  $\mathcal{CO}(Y)$   of characteristic maps $Y \to S$; this set can be given the compact-open topology. This topology is non-Hausdorff of course. 
