You need a cat to always have access to a room, so the door should be left open, but the door should also be able to be closed if you want to.
It was so easy, that I actually hesitated to make an Instructable.
Step 1: What You Need and How to Do It.
1 rubber band.
Fasten the two nails in the upper corner of the door post and connect a rubber band between them. The door will now always stay partly open.