Janet, what do you mean when you say to 'weight' the cabbage?
It needs to be weighted down to keep it submerged in the brine. I do the brining in a container large enough to hold the cabbage and enough brine to just cover it, wide enough that I can then stack dessert or salad plates and bowls on top of the cabbage to keep the cabbage covered by the liquid. Note that during the hours of brining, a lot of the water within the cabbage will be leaving the cabbage, so you end up with reduced volume of wilted, salty cabbage and a lot more fluid in your container - so plan accordingly when selecting bowl size to avoid overflow.
This is DIFFERENT from weighting when making sauerkraut or dill pickles, where airtightness is also a factor. For this initial brine, nothing wrong with leaving everything unlidded and open to the air.