At least keep them out of your bedroom at all times. Cat dander and cat saliva are highly allergic. Dander and saliva stay in the house for months after a cat is removed. If dust bothers you, put special covers over your mattress and bed pillows. This washing kills many of the dust mites that live in the bedding.
Dust mites can trigger asthma attacks.
Have someone else vacuum your carpets once or twice a week. This cleaning cuts down on house dust and kills dust mites that live in the carpets. You shouldn't go into the room for several hours after the vacuuming is done. If you can, take out the carpeting altogether. Bare floors are easier to keep clean.