How do I make my dome light fade away when you shut the door?

Well.........if you just stick a big huge cap across the lamp, it will take a BIG cap to provide any delay, but the PROBLEM is that when you OPEN the door, the cap will act like a "dead short" during it's charging cycle, and will be not only hard on your door switches, but may blow the fuse, too.

Thats what I love about this site, no matter what is said, someone disagrees.

Capacitors are what is used to make those dimming "auto" features work, sometimes, as far as the specifics of setting it up or engineering it properly, I don't care enough about it to invest the time. I was just giving the guy a place to start.

BTW I seriously doubt you know what "big" means relative to size of the bulb in the dome and associated wiring.