In most case, the definition of a variety over a field $k$ at least requires that being "of finite type" and being "separated". It has no question for me that being of finite type, since we always like finite.
I donot know the reason why we require being "separated" to a variety?
There is a reason since a scheme over a field $k$ being separated will have the property that the intersection of two affine open sets is still affine open set.
Are there any other acceptant reasons?
Thanks a lot.