Are thinking of a duvet? That is like a comforter, inside a large cloth bag which is known as a duvet cover. If you use a duvet cover, you don't need a top sheet, but you remove the cover and wash it as often as you would wash a top sheet.
If you put a bag over the mattress--not that I've ever seen such a thing that takes the place of a sheet--then you would have to remove the mattress from the bag to wash it as often as you wash a sheet. That seems like a lot of work. I don't know about you, but I can't lift a mattress. Buy fitted sheets, the right size for the mattress, and I don't know why you'd have trouble with them.
It's called a mattress bag.. The thing you put on top of it is a ducet. Nevertheless, you should always have a mattress pad, and then sheets, and ducet if you want it or your bed clothing. We spend a third of our lives in bed, and it should be a good experience.
There is this thing I've seen in the past that is similar to a sheet. It is like a bag you put a mattress inside only it's made of cloth. I've got a new mattress coming in so I'd like to use one of these since I have no luck with actual sheets. The bed is a full size 10" thick memory foam mattress. Any ideas?