Now if this place was "off campus housing" for college students, chances are that one of the students would fit a bed in that walk-in closet (assuming a bed would in fact fit flat in there). Most college students hate having to share a sleeping room; a separate (no matter how small and crampt) sleeping room is still highly prized. Doesn't matter if there's no window; the ability to shut the door to keep noise and light out is all that's wanted. So yes it should get an AFCI.
