Questions:
1) What is the area of the closet?
2) Is this in the US? If so the International Building Code (IBC) is most likely the applicable code.
I'm assuming that since it is a closet, the requirement for exit illumination in the IBC are probably not applicable. IBC section 1006.1 does require exit illumination in the means of egress. If it's a closet, its not a part of the egress path and illumination is not required.
If your speaking to exit signs IBC section 1011.1, exception 1 does not require them when the room only requires one exit or exit access opening.