Both men and women should close the toilet lid before flushing. If you flush with the lid up, a polluted plume of bacteria and water vapour erupts out of the flushing toilet bowl. The polluted water particles float for a few hours around your bathroom before they all land. Some of them will land on your towels and tooth brush.
Google “toilet flush bacteria.”
Not that I don’t like the toidy lid down, but I think MythBusters shot that one down.
There was an extrapolation of how many times the toilet is used in the down position and in the up position. Far more uses down (and in my family, the guys all go outside to pee on a tree, so it’s like 0 uses up).
That being said, our facilities are not in the same room with our sink. However, I will give this info to Mr. Goldfinch. :-)