Is a smoke detector required outside a bedroom if there is no door separating the bedroom from the living room?
In my opinion, yes. The 2006 IRC section R313.2 says "1. In each sleeping room. 2. Outside each sleeping area in the immediate vicinity of the bedrooms. 3. On each additional story of the dwelling."
(No mention of doors) NFPA 72 section 11.5.1
Required Detection- says (1) In all sleeping rooms and guest rooms. (2) Outside of each separate dwelling unit sleeping area, within 21ft of any door to a sleeping room, the distance measured along a path of travel. (3) On every level of a dwelling unit, including basements.
An opening may be defined as a doorway, depending on who is making the call. IMO the code doesn't care if there is a door because NFPA 72 also requires that a wall mounted dectector not be farther than 12" from the adjoining ceiling surface.
To my knowledge, the NEC has no requirements for Residential smoke detectors. So it would depend on what codes are adopted in your jurisdiction.
In our jurisdiction if the room has a closet and an emergency escape window, then it is a bedroom.