Why not just have arrows in the category of opens represent coverings instead of inclusions?


It seems to me like both conventions (whether presheaves are co/contra and which of the two dual orderings to use to represent spaces as thin categories) are pretty arbitrary, and switching them would simplify notation.


Is it a topology-as-logic issue, where we want to be able to regard meets as conjunctions (a point satisfies $a \wedge b$ when it satisfies both opens $a,b$) etc.?


What, if anything, am I missing?



Why are (pre)sheaves defined as contravariant functors? Why not just reverse the arrows in the first place?
The category of open subsets of a space $X$ is a subcategory of the category of spaces $Y \to X$ over $X$, so the usual direction of arrows is natural in this sense. 


Constructing the category of presheaves $\text{Psh}(C)$ on a category $C$ is a covariant construction. Moreover, it is the [free cocompletion](http://ncatlab.org/nlab/show/free+cocompletion) of $C$. So the category of presheaves over $X$ is a certain completion of a certain subcategory of the category of spaces over $X$, which should give you some indication of the relationship between sheaves and [étalé spaces](http://en.wikipedia.org/wiki/Sheaf_(mathematics)#The_.C3.A9tal.C3.A9_space_of_a_sheaf). In somewhat more detail, there is an adjunction between the category of spaces over $X$ and the category of presheaves over $X$, and this adjunction restricts canonically to an equivalence of categories between étalé spaces and sheaves. 


