Sounds like the light in my refrigerator. At least I THINK it goes off when the door is shut. Who are you keeping cold in this closet?

I actually did solve your problem once when I was a kid. I took a "open when pressed" push button and mounted it thru the light switch plate of my bedroom closet. The light was wired to this switch and would come on when the button was released and go off when the button was pressed. To the door I mounted a wood yardstick (I was a kid, remember) that extended past the door's edge and would press the button when the door was closed. Worked great. Door opened INTO the closet though and switch was right next to the door frame. Might be harder to screw a yardstick on the outside of a door and get away with it.

Just a thought.

Dave